WikiWaves - User contributions [en]

Boundary Element Method for a Fixed Body in Finite Depth

2009-11-29T23:06:34Z

Cfitzgerald: /* Singularity Expansion Method */

{{complete pages}}

== Introduction ==

We show how to solve for the wave scattering by a rigid body in constant [[Finite Depth]] using
the [[:Category:Boundary Element Method|Boundary Element Method]]. The method can be modified
to account for a body which can move and this is described in
[[Boundary Element Method for a Floating Body in Finite Depth]]

== Equations ==

{{boundary value problem for a fixed body}}

{{incident plane wave}}

{{sommerfeld radiation condition two dimensions}}

== Solution Method ==

We divide the domain into three regions, <math>x<-l</math>,
<math>x>r</math>, and <math>-l<x<r</math> so that the body surface is entirely in the
finite region.

=== Solution in the finite region ===

We use the [[:Category:Boundary Element Method|Boundary Element Method]]
in the finite region.

===Solution in the Semi-infinite Domains===

We now solve Laplace's equation in the semi-infinite domains. First consider
the domain on the left so that <math>\Omega = \left\{ x<-l,\;-h\leq z\leq 0\right\} </math>
The equations are
{{standard linear wave scattering equations without body condition}}
We have the following explicit boundary conditions
<center><math>
\phi =f\left( z\right) ,\;\;x=-\,l
</math></center>
and
<center><math>
\lim\limits_{x\rightarrow -\infty }\phi \left( x,z\right) = \phi_0
e^{-k_{0}x}
+R\phi_0
e^{k_{0}x},
</math></center>
where <math>f\left( z\right) </math> is
an arbitrary continuous function<math>.</math> Our aim is to find the outward normal
derivative of the potential on <math>x=-l</math> as a function of <math>\tilde{\phi}\left(
z\right) </math>. <math>\phi_0</math> and <math>k_0</math> will be defined shortly
when we separate variables, but are equivalent to the [[Sommerfeld Radiation Condition]].

Since the water depth is constant in these regions we can solve Laplace's
equation by separation of variables.

{{separation of variables for a free surface}}

=== Expansion of the potential ===

This gives us the following expression for the potential in the region <math>
\Omega </math>
<center><math>
\phi \left( x,z\right) = \phi_0(z)e^{-k_0 x}
+\sum_{m=0}^{\infty } a_m \phi _{m}\left( z\right)
e^{k_{m}\left( x+l\right) }
</math></center>
where <math>a_m</math> can be found from <math>\left. \phi \right| _{x=-l} = f(z)</math>
from the orthogonality of the vertical eigenfunctions. Therefore
<center><math>
a_m +\delta_{m0}e^{k_0 l} = \frac{1}{A_m} \int_{-h}^{0} f(z) \phi_m(z) \mathrm{d}z
</math></center>

The normal derivative of potential (with the normal outward from the interior region) we obtain
<center><math>
\partial_n\left. \phi \right| _{x=-l}
= -\partial_x \left. \phi \right| _{x=-l}
= k_0 \phi_0(z)e^{k_0 l}
- \sum_{m=0}^{\infty } k_m a_m \phi _{m}\left( z\right)
</math></center>

If we define <math>\mathbf{Q}</math> as
<center><math>
\mathbf{Q} \left[f\left( z\right) \right]=
-\sum_{m=0}^{\infty } \frac{k_{m}}{A_m} \left\{ \int_{-h}^{0} f\left( s\right) \phi
_{m}\left( s\right) \mathrm{d}s \right\} \phi _{m}\left( z\right)
</math></center>
we can write
<center><math>
\partial_n \left. \phi \right| _{x=-l} =\mathbf{Q} \left[ f(z) \right] + 2k_0 \phi_0e^{k_0l},
</math></center>

Similarly, we now consider the potential in the region
<math>\Omega = \left\{ x>r,\;-h\leq z\leq 0\right\} </math>
which satisfies exactly the equations as before except the boundary
condition is
<center><math>
\lim\limits_{x\rightarrow \infty }\phi \left( x,z\right) = \phi_0
T e^{-k_{0}x}
</math></center>
We solve again by separation of variables and obtain
<center><math>
\left. \partial_n \phi \right| _{x=r}=
\left. \partial_x \phi \right| _{x=r}=\mathbf{Q}\left[f\left( z\right)\right] ,
</math></center>
where the outward normal is with respect to the inner domain as before. Note that, if
the depth is different on each side then the matrix <math>\mathbf{Q}</math> will be different
(since we need to use a different depth solving the [[Dispersion Relation for a Free Surface]] etc.)

== Boundary Element Method ==

We have reduced the problem to Laplace's equation in a finite domain subject
to certain boundary conditions. These boundary
conditions give the outward normal derivative of the potential as a function
of the potential but this is not always a point-wise condition; on some
boundaries it is given by an integral equation. We must solve both Laplace's
equation and the integral equations numerically. We will solve Laplace's
equation by the [[:Category:Boundary Element Method|Boundary Element Method]] and the integral equations by
numerical integration. However, the same discretisation of the boundary will
be used for both numerical solutions.
We solve Laplace's equation by a modified constant panel method
which reduces it to the following matrix equation
<center><math>
\frac{1}{2}\vec{\phi}=\partial_n\mathbf{G}\vec{\phi}-\mathbf{G}\partial_n\vec{\phi}.
</math></center>
where <math>\vec{\phi}\mathcal{\ }</math>and <math>\partial_n\vec{\phi}</math>
are vectors which approximate the potential and its normal derivative
around the boundary <math>\partial \Omega </math>, and <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>
are matrices corresponding to the Green function and the outward normal
derivative of the Green function respectively. The method used to calculate
the elements of the matrices <math>\mathbf{G}</math> and <math>\partial_n\mathbf{G}</math> can be
found in [[:Category:Boundary Element Method|Boundary Element Method]].

The outward normal derivative of the potential, <math>\partial_n\vec{\phi},</math> and the
potential, <math>\vec{\phi},</math> are related by the conditions on the boundary <math>
\partial \Omega </math>. This can be expressed as
<center><math>
\partial_n\vec{\phi}=\mathbf{A}\,\vec{\phi}-\vec{f},
</math></center>
where <math>\mathbf{A}</math> is the block diagonal matrix given by
<center><math>
\mathbf{A}\mathbb{=}\left[
\begin{matrix}
\mathbf{Q} & & & \\
& \mathbf{Q} & & \\
& & \alpha \,\mathbf{I} & \\
& & & 0
\end{matrix}
\right] ,
</math></center>
where we have separated the boundaries to the semi-infinite domains, the free surface and the
sea floor and body boundary.
<math>\mathbf{Q}</math> is a matrix
approximation of the integral operator of the same name and <math>\vec{f}</math> is
the vector
<center><math>
\vec{f}=\left[
\begin{matrix}
-2k_{0}\phi_0 \,e^{k_{0}l } \\
0 \\
\vdots \\
0
\end{matrix}
\right] .
</math></center>

We obtain the following matrix equation for the
potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
which can be solved straightforwardly. The reflection and transmission
coefficients are calculated from the solution.

===Numerical Calculation of <math>\mathbf{Q}</math> ===

We begin by truncating to a finite number (<math>N</math>) of
evanescent modes,
<center><math>
\mathbf{Q}\left[ f \right] =-\sum_{m=0}^{N}k_{m}\int_{-h}^{0}
f\left( s\right) \phi_{m}\left( s\right) \mathrm{d}s
\frac{\phi _{m} \left( z\right)}{A_m} .
</math></center>
We calculate the integral
with the same panels as we used to approximate the integrals of the Green
function and its normal derivative . Similarly, we
assume that <math>f(s) \,</math> is constant over each panel and integrate <math>\phi
_{m}\left( s\right) </math> exactly. This gives us the
following matrix factorisation of <math>\mathbf{Q}</math>
<center><math>
\mathbf{Q}[f]=\mathbf{S}\,\mathbf{R}\,[f].
</math></center>
The components of the matrices <math>\mathbf{S}</math> and <math>\mathbf{R}</math> are
<center><math>
s_{jm} = -k_m\phi _{m}\left( z_{j}\right) ,
</math></center>
<center><math>
r_{mj} = \frac{1}{A_m} \int_{z_{j}-\Delta x/2}^{z_{j}+\Delta x/2}\phi
_{m}\left( s\right) \mathrm{d}s
</math></center>

=== Reflection and Transmission Coefficients ===

If we multiply the potential at the left (after subtracting the incident wave) and the right by
<math>\mathbf{R}</math> we can calculate the coefficients in the eigenfunction expansion, and
hence determine the reflection and transmission coefficient.
where <math>z_{j}</math> is the value of the <math>z</math> coordinate in the centre of the panel
and <math>\Delta x</math> is the panel length.

=== Singularity Expansion Method ===
The matrix equation for the potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
can be rewritten as
<center><math>
\mathbf{M}(k) \mathbf{x}=\mathbf{F}(k)
</math></center>
where <math>\mathbf{x}</math> represents the unknown potential values in the new notation. Resonant frequencies (or scattering frequencies) correspond to zero eigenvalues of the matrix <math>\mathbf{M}(k)</math> for a set of discrete points <math>k=k_p</math> in the upper half of the complex plane (due to the time dependence). Since
<center><math>
\mathbf{x}=\left( \mathbf{M}(k)\right)^{-1}\mathbf{F}(k)
</math></center>
then <math>\mathbf{x}</math> will be singular for these scattering frequencies. Each scattering frequency, corresponding to a zero eigenvalue of <math>\mathbf{M}(k)</math> will possess a corresponding eigenvector <math>\mathbf{u}_{k_p}</math> satisfying
<center><math>
\mathbf{M}(k_p)\mathbf{u}_{k_p}=0
</math></center>
and also a related conjugate transpose eigenvector <math>\mathbf{u}^{*}_{k_p}</math> satisfying
<center><math>
\mathbf{M}(k_p)^{A}\mathbf{u}^*_{k_p}=0.
</math></center>
In the vicinity of a scattering frequency it can be shown that the vector of unknowns satisfies
<center><math>
\mathbf{x}(k)\approx \frac{\langle \mathbf{F}(k_p),\mathbf{u}^*_{k^*_p} \rangle}{\langle\mathbf{u}_{k_p},\mathbf{M}'(k_p)\mathbf{u}^*_{k^*_p}\rangle}\frac{1}{k-k_p}
</math></center>
and we define a corresponding mode shape as
<center><math>
\zeta_{k_p}(x,k_p) =\frac{\langle \mathbf{F}(k_p),\mathbf{u}^*_{k^*_p} \rangle}{\langle\mathbf{u}_{k_p},\mathbf{M}'(k_p)\mathbf{u}^*_{k^*_p}\rangle}U(x)
</math></center>
where <math>U(x)</math> is obtained

== Matlab Code ==

{{fixed body 2d bem code}}

=== Additional code ===

This program requires
* {{free surface dispersion equation code}}
* {{boundary element code}}

[[Category:Boundary Element Method]]
[[Category:Eigenfunction Matching Method]]

Boundary Element Method for a Fixed Body in Finite Depth

2009-11-29T22:59:11Z

Cfitzgerald: /* Singularity Expansion Method */

{{complete pages}}

== Introduction ==

We show how to solve for the wave scattering by a rigid body in constant [[Finite Depth]] using
the [[:Category:Boundary Element Method|Boundary Element Method]]. The method can be modified
to account for a body which can move and this is described in
[[Boundary Element Method for a Floating Body in Finite Depth]]

== Equations ==

{{boundary value problem for a fixed body}}

{{incident plane wave}}

{{sommerfeld radiation condition two dimensions}}

== Solution Method ==

We divide the domain into three regions, <math>x<-l</math>,
<math>x>r</math>, and <math>-l<x<r</math> so that the body surface is entirely in the
finite region.

=== Solution in the finite region ===

We use the [[:Category:Boundary Element Method|Boundary Element Method]]
in the finite region.

===Solution in the Semi-infinite Domains===

We now solve Laplace's equation in the semi-infinite domains. First consider
the domain on the left so that <math>\Omega = \left\{ x<-l,\;-h\leq z\leq 0\right\} </math>
The equations are
{{standard linear wave scattering equations without body condition}}
We have the following explicit boundary conditions
<center><math>
\phi =f\left( z\right) ,\;\;x=-\,l
</math></center>
and
<center><math>
\lim\limits_{x\rightarrow -\infty }\phi \left( x,z\right) = \phi_0
e^{-k_{0}x}
+R\phi_0
e^{k_{0}x},
</math></center>
where <math>f\left( z\right) </math> is
an arbitrary continuous function<math>.</math> Our aim is to find the outward normal
derivative of the potential on <math>x=-l</math> as a function of <math>\tilde{\phi}\left(
z\right) </math>. <math>\phi_0</math> and <math>k_0</math> will be defined shortly
when we separate variables, but are equivalent to the [[Sommerfeld Radiation Condition]].

Since the water depth is constant in these regions we can solve Laplace's
equation by separation of variables.

{{separation of variables for a free surface}}

=== Expansion of the potential ===

This gives us the following expression for the potential in the region <math>
\Omega </math>
<center><math>
\phi \left( x,z\right) = \phi_0(z)e^{-k_0 x}
+\sum_{m=0}^{\infty } a_m \phi _{m}\left( z\right)
e^{k_{m}\left( x+l\right) }
</math></center>
where <math>a_m</math> can be found from <math>\left. \phi \right| _{x=-l} = f(z)</math>
from the orthogonality of the vertical eigenfunctions. Therefore
<center><math>
a_m +\delta_{m0}e^{k_0 l} = \frac{1}{A_m} \int_{-h}^{0} f(z) \phi_m(z) \mathrm{d}z
</math></center>

The normal derivative of potential (with the normal outward from the interior region) we obtain
<center><math>
\partial_n\left. \phi \right| _{x=-l}
= -\partial_x \left. \phi \right| _{x=-l}
= k_0 \phi_0(z)e^{k_0 l}
- \sum_{m=0}^{\infty } k_m a_m \phi _{m}\left( z\right)
</math></center>

If we define <math>\mathbf{Q}</math> as
<center><math>
\mathbf{Q} \left[f\left( z\right) \right]=
-\sum_{m=0}^{\infty } \frac{k_{m}}{A_m} \left\{ \int_{-h}^{0} f\left( s\right) \phi
_{m}\left( s\right) \mathrm{d}s \right\} \phi _{m}\left( z\right)
</math></center>
we can write
<center><math>
\partial_n \left. \phi \right| _{x=-l} =\mathbf{Q} \left[ f(z) \right] + 2k_0 \phi_0e^{k_0l},
</math></center>

Similarly, we now consider the potential in the region
<math>\Omega = \left\{ x>r,\;-h\leq z\leq 0\right\} </math>
which satisfies exactly the equations as before except the boundary
condition is
<center><math>
\lim\limits_{x\rightarrow \infty }\phi \left( x,z\right) = \phi_0
T e^{-k_{0}x}
</math></center>
We solve again by separation of variables and obtain
<center><math>
\left. \partial_n \phi \right| _{x=r}=
\left. \partial_x \phi \right| _{x=r}=\mathbf{Q}\left[f\left( z\right)\right] ,
</math></center>
where the outward normal is with respect to the inner domain as before. Note that, if
the depth is different on each side then the matrix <math>\mathbf{Q}</math> will be different
(since we need to use a different depth solving the [[Dispersion Relation for a Free Surface]] etc.)

== Boundary Element Method ==

We have reduced the problem to Laplace's equation in a finite domain subject
to certain boundary conditions. These boundary
conditions give the outward normal derivative of the potential as a function
of the potential but this is not always a point-wise condition; on some
boundaries it is given by an integral equation. We must solve both Laplace's
equation and the integral equations numerically. We will solve Laplace's
equation by the [[:Category:Boundary Element Method|Boundary Element Method]] and the integral equations by
numerical integration. However, the same discretisation of the boundary will
be used for both numerical solutions.
We solve Laplace's equation by a modified constant panel method
which reduces it to the following matrix equation
<center><math>
\frac{1}{2}\vec{\phi}=\partial_n\mathbf{G}\vec{\phi}-\mathbf{G}\partial_n\vec{\phi}.
</math></center>
where <math>\vec{\phi}\mathcal{\ }</math>and <math>\partial_n\vec{\phi}</math>
are vectors which approximate the potential and its normal derivative
around the boundary <math>\partial \Omega </math>, and <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>
are matrices corresponding to the Green function and the outward normal
derivative of the Green function respectively. The method used to calculate
the elements of the matrices <math>\mathbf{G}</math> and <math>\partial_n\mathbf{G}</math> can be
found in [[:Category:Boundary Element Method|Boundary Element Method]].

The outward normal derivative of the potential, <math>\partial_n\vec{\phi},</math> and the
potential, <math>\vec{\phi},</math> are related by the conditions on the boundary <math>
\partial \Omega </math>. This can be expressed as
<center><math>
\partial_n\vec{\phi}=\mathbf{A}\,\vec{\phi}-\vec{f},
</math></center>
where <math>\mathbf{A}</math> is the block diagonal matrix given by
<center><math>
\mathbf{A}\mathbb{=}\left[
\begin{matrix}
\mathbf{Q} & & & \\
& \mathbf{Q} & & \\
& & \alpha \,\mathbf{I} & \\
& & & 0
\end{matrix}
\right] ,
</math></center>
where we have separated the boundaries to the semi-infinite domains, the free surface and the
sea floor and body boundary.
<math>\mathbf{Q}</math> is a matrix
approximation of the integral operator of the same name and <math>\vec{f}</math> is
the vector
<center><math>
\vec{f}=\left[
\begin{matrix}
-2k_{0}\phi_0 \,e^{k_{0}l } \\
0 \\
\vdots \\
0
\end{matrix}
\right] .
</math></center>

We obtain the following matrix equation for the
potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
which can be solved straightforwardly. The reflection and transmission
coefficients are calculated from the solution.

===Numerical Calculation of <math>\mathbf{Q}</math> ===

We begin by truncating to a finite number (<math>N</math>) of
evanescent modes,
<center><math>
\mathbf{Q}\left[ f \right] =-\sum_{m=0}^{N}k_{m}\int_{-h}^{0}
f\left( s\right) \phi_{m}\left( s\right) \mathrm{d}s
\frac{\phi _{m} \left( z\right)}{A_m} .
</math></center>
We calculate the integral
with the same panels as we used to approximate the integrals of the Green
function and its normal derivative . Similarly, we
assume that <math>f(s) \,</math> is constant over each panel and integrate <math>\phi
_{m}\left( s\right) </math> exactly. This gives us the
following matrix factorisation of <math>\mathbf{Q}</math>
<center><math>
\mathbf{Q}[f]=\mathbf{S}\,\mathbf{R}\,[f].
</math></center>
The components of the matrices <math>\mathbf{S}</math> and <math>\mathbf{R}</math> are
<center><math>
s_{jm} = -k_m\phi _{m}\left( z_{j}\right) ,
</math></center>
<center><math>
r_{mj} = \frac{1}{A_m} \int_{z_{j}-\Delta x/2}^{z_{j}+\Delta x/2}\phi
_{m}\left( s\right) \mathrm{d}s
</math></center>

=== Reflection and Transmission Coefficients ===

If we multiply the potential at the left (after subtracting the incident wave) and the right by
<math>\mathbf{R}</math> we can calculate the coefficients in the eigenfunction expansion, and
hence determine the reflection and transmission coefficient.
where <math>z_{j}</math> is the value of the <math>z</math> coordinate in the centre of the panel
and <math>\Delta x</math> is the panel length.

=== Singularity Expansion Method ===
The matrix equation for the potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
can be rewritten as
<center><math>
\mathbf{M}(k) \mathbf{x}=\mathbf{F}(k)
</math></center>
where <math>\mathbf{x}</math> represents the unknown potential values in the new notation. Resonant frequencies (or scattering frequencies) correspond to zero eigenvalues of the matrix <math>\mathbf{M}(k)</math> for a set of discrete points <math>k=k_p</math> in the upper half of the complex plane (due to the time dependence). Since
<center><math>
\mathbf{x}=\left( \mathbf{M}(k)\right)^{-1}\mathbf{F}(k)
</math></center>
then <math>\mathbf{x}</math> will be singular for these scattering frequencies. Each scattering frequency, corresponding to a zero eigenvalue of <math>\mathbf{M}(k)</math> will possess a corresponding eigenvector <math>\mathbf{u}_{k_p}</math> satisfying
<center><math>
\mathbf{M}(k_p)\mathbf{u}_{k_p}=0
</math></center>
and also a related conjugate transpose eigenvector <math>\mathbf{u}^{*}_{k_p}</math> satisfying
<center><math>
\mathbf{M}(k_p)^{A}\mathbf{u}^*_{k_p}=0.
</math></center>
In the vicinity of a scattering frequency it can be shown that the vector of unknowns satisfies
<center><math>
\mathbf{x}(k)\approx \frac{\langle \mathbf{F}(k_p),\mathbf{u}^*_{k^*_p} \rangle}{\langle\mathbf{u}_{k_p},\mathbf{M}'(k_p)\mathbf{u}^*_{k^*_p}\rangle}\frac{1}{k-k_p}
</math></center>
and we define a corresponding mode shape as
<center><math>
\zeta_{k_p}(x,k_p) =\frac{\langle \mathbf{F}(k_p),\mathbf{u}^*_{k^*_p} \rangle}{\langle\mathbf{u}_{k_p},\mathbf{M}'(k)\mathbf{u}^*_{k^*_p}\rangle}U(x)
</math></center>
where <math>U(x)</math> is obtained

== Matlab Code ==

{{fixed body 2d bem code}}

=== Additional code ===

This program requires
* {{free surface dispersion equation code}}
* {{boundary element code}}

[[Category:Boundary Element Method]]
[[Category:Eigenfunction Matching Method]]

Boundary Element Method for a Fixed Body in Finite Depth

2009-11-29T22:58:33Z

Cfitzgerald: /* Singularity Expansion Method */

{{complete pages}}

== Introduction ==

We show how to solve for the wave scattering by a rigid body in constant [[Finite Depth]] using
the [[:Category:Boundary Element Method|Boundary Element Method]]. The method can be modified
to account for a body which can move and this is described in
[[Boundary Element Method for a Floating Body in Finite Depth]]

== Equations ==

{{boundary value problem for a fixed body}}

{{incident plane wave}}

{{sommerfeld radiation condition two dimensions}}

== Solution Method ==

We divide the domain into three regions, <math>x<-l</math>,
<math>x>r</math>, and <math>-l<x<r</math> so that the body surface is entirely in the
finite region.

=== Solution in the finite region ===

We use the [[:Category:Boundary Element Method|Boundary Element Method]]
in the finite region.

===Solution in the Semi-infinite Domains===

We now solve Laplace's equation in the semi-infinite domains. First consider
the domain on the left so that <math>\Omega = \left\{ x<-l,\;-h\leq z\leq 0\right\} </math>
The equations are
{{standard linear wave scattering equations without body condition}}
We have the following explicit boundary conditions
<center><math>
\phi =f\left( z\right) ,\;\;x=-\,l
</math></center>
and
<center><math>
\lim\limits_{x\rightarrow -\infty }\phi \left( x,z\right) = \phi_0
e^{-k_{0}x}
+R\phi_0
e^{k_{0}x},
</math></center>
where <math>f\left( z\right) </math> is
an arbitrary continuous function<math>.</math> Our aim is to find the outward normal
derivative of the potential on <math>x=-l</math> as a function of <math>\tilde{\phi}\left(
z\right) </math>. <math>\phi_0</math> and <math>k_0</math> will be defined shortly
when we separate variables, but are equivalent to the [[Sommerfeld Radiation Condition]].

Since the water depth is constant in these regions we can solve Laplace's
equation by separation of variables.

{{separation of variables for a free surface}}

=== Expansion of the potential ===

This gives us the following expression for the potential in the region <math>
\Omega </math>
<center><math>
\phi \left( x,z\right) = \phi_0(z)e^{-k_0 x}
+\sum_{m=0}^{\infty } a_m \phi _{m}\left( z\right)
e^{k_{m}\left( x+l\right) }
</math></center>
where <math>a_m</math> can be found from <math>\left. \phi \right| _{x=-l} = f(z)</math>
from the orthogonality of the vertical eigenfunctions. Therefore
<center><math>
a_m +\delta_{m0}e^{k_0 l} = \frac{1}{A_m} \int_{-h}^{0} f(z) \phi_m(z) \mathrm{d}z
</math></center>

The normal derivative of potential (with the normal outward from the interior region) we obtain
<center><math>
\partial_n\left. \phi \right| _{x=-l}
= -\partial_x \left. \phi \right| _{x=-l}
= k_0 \phi_0(z)e^{k_0 l}
- \sum_{m=0}^{\infty } k_m a_m \phi _{m}\left( z\right)
</math></center>

If we define <math>\mathbf{Q}</math> as
<center><math>
\mathbf{Q} \left[f\left( z\right) \right]=
-\sum_{m=0}^{\infty } \frac{k_{m}}{A_m} \left\{ \int_{-h}^{0} f\left( s\right) \phi
_{m}\left( s\right) \mathrm{d}s \right\} \phi _{m}\left( z\right)
</math></center>
we can write
<center><math>
\partial_n \left. \phi \right| _{x=-l} =\mathbf{Q} \left[ f(z) \right] + 2k_0 \phi_0e^{k_0l},
</math></center>

Similarly, we now consider the potential in the region
<math>\Omega = \left\{ x>r,\;-h\leq z\leq 0\right\} </math>
which satisfies exactly the equations as before except the boundary
condition is
<center><math>
\lim\limits_{x\rightarrow \infty }\phi \left( x,z\right) = \phi_0
T e^{-k_{0}x}
</math></center>
We solve again by separation of variables and obtain
<center><math>
\left. \partial_n \phi \right| _{x=r}=
\left. \partial_x \phi \right| _{x=r}=\mathbf{Q}\left[f\left( z\right)\right] ,
</math></center>
where the outward normal is with respect to the inner domain as before. Note that, if
the depth is different on each side then the matrix <math>\mathbf{Q}</math> will be different
(since we need to use a different depth solving the [[Dispersion Relation for a Free Surface]] etc.)

== Boundary Element Method ==

We have reduced the problem to Laplace's equation in a finite domain subject
to certain boundary conditions. These boundary
conditions give the outward normal derivative of the potential as a function
of the potential but this is not always a point-wise condition; on some
boundaries it is given by an integral equation. We must solve both Laplace's
equation and the integral equations numerically. We will solve Laplace's
equation by the [[:Category:Boundary Element Method|Boundary Element Method]] and the integral equations by
numerical integration. However, the same discretisation of the boundary will
be used for both numerical solutions.
We solve Laplace's equation by a modified constant panel method
which reduces it to the following matrix equation
<center><math>
\frac{1}{2}\vec{\phi}=\partial_n\mathbf{G}\vec{\phi}-\mathbf{G}\partial_n\vec{\phi}.
</math></center>
where <math>\vec{\phi}\mathcal{\ }</math>and <math>\partial_n\vec{\phi}</math>
are vectors which approximate the potential and its normal derivative
around the boundary <math>\partial \Omega </math>, and <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>
are matrices corresponding to the Green function and the outward normal
derivative of the Green function respectively. The method used to calculate
the elements of the matrices <math>\mathbf{G}</math> and <math>\partial_n\mathbf{G}</math> can be
found in [[:Category:Boundary Element Method|Boundary Element Method]].

The outward normal derivative of the potential, <math>\partial_n\vec{\phi},</math> and the
potential, <math>\vec{\phi},</math> are related by the conditions on the boundary <math>
\partial \Omega </math>. This can be expressed as
<center><math>
\partial_n\vec{\phi}=\mathbf{A}\,\vec{\phi}-\vec{f},
</math></center>
where <math>\mathbf{A}</math> is the block diagonal matrix given by
<center><math>
\mathbf{A}\mathbb{=}\left[
\begin{matrix}
\mathbf{Q} & & & \\
& \mathbf{Q} & & \\
& & \alpha \,\mathbf{I} & \\
& & & 0
\end{matrix}
\right] ,
</math></center>
where we have separated the boundaries to the semi-infinite domains, the free surface and the
sea floor and body boundary.
<math>\mathbf{Q}</math> is a matrix
approximation of the integral operator of the same name and <math>\vec{f}</math> is
the vector
<center><math>
\vec{f}=\left[
\begin{matrix}
-2k_{0}\phi_0 \,e^{k_{0}l } \\
0 \\
\vdots \\
0
\end{matrix}
\right] .
</math></center>

We obtain the following matrix equation for the
potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
which can be solved straightforwardly. The reflection and transmission
coefficients are calculated from the solution.

===Numerical Calculation of <math>\mathbf{Q}</math> ===

We begin by truncating to a finite number (<math>N</math>) of
evanescent modes,
<center><math>
\mathbf{Q}\left[ f \right] =-\sum_{m=0}^{N}k_{m}\int_{-h}^{0}
f\left( s\right) \phi_{m}\left( s\right) \mathrm{d}s
\frac{\phi _{m} \left( z\right)}{A_m} .
</math></center>
We calculate the integral
with the same panels as we used to approximate the integrals of the Green
function and its normal derivative . Similarly, we
assume that <math>f(s) \,</math> is constant over each panel and integrate <math>\phi
_{m}\left( s\right) </math> exactly. This gives us the
following matrix factorisation of <math>\mathbf{Q}</math>
<center><math>
\mathbf{Q}[f]=\mathbf{S}\,\mathbf{R}\,[f].
</math></center>
The components of the matrices <math>\mathbf{S}</math> and <math>\mathbf{R}</math> are
<center><math>
s_{jm} = -k_m\phi _{m}\left( z_{j}\right) ,
</math></center>
<center><math>
r_{mj} = \frac{1}{A_m} \int_{z_{j}-\Delta x/2}^{z_{j}+\Delta x/2}\phi
_{m}\left( s\right) \mathrm{d}s
</math></center>

=== Reflection and Transmission Coefficients ===

If we multiply the potential at the left (after subtracting the incident wave) and the right by
<math>\mathbf{R}</math> we can calculate the coefficients in the eigenfunction expansion, and
hence determine the reflection and transmission coefficient.
where <math>z_{j}</math> is the value of the <math>z</math> coordinate in the centre of the panel
and <math>\Delta x</math> is the panel length.

=== Singularity Expansion Method ===
The matrix equation for the potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
can be rewritten as
<center><math>
\mathbf{M}(k) \mathbf{x}=\mathbf{F}(k)
</math></center>
where <math>\mathbf{x}</math> represents the unknown potential values in the new notation. Resonant frequencies (or scattering frequencies) correspond to zero eigenvalues of the matrix <math>\mathbf{M}(k)</math> for a set of discrete points <math>k=k_p</math> in the upper half of the complex plane (due to the time dependence). Since
<center><math>
\mathbf{x}=\left( \mathbf{M}(k)\right)^{-1}\mathbf{F}(k)
</math></center>
then <math>\mathbf{x}</math> will be singular for these scattering frequencies. Each scattering frequency, corresponding to a zero eigenvalue of <math>\mathbf{M}(k)</math> will possess a corresponding eigenvector <math>\mathbf{u}_{k_p}</math> satisfying
<center><math>
\mathbf{M}(k_p)\mathbf{u}_{k_p}=0
</math></center>
and also a related conjugate transpose eigenvector <math>\mathbf{u}^{*}_{k_p}</math> satisfying
<center><math>
\mathbf{M}(k_p)^{A}\mathbf{u}^*_{k_p}=0.
</math></center>
In the vicinity of a scattering frequency it can be shown that the vector of unknowns satisfies
<center><math>
\mathbf{x}(k)\approx \frac{\langle \mathbf{F}(k_p),\mathbf{u}^*_{k^*_p} \rangle}{\langle\mathbf{u}_{k_p},\mathbf{M}'(k_p)\mathbf{u}^*_{k^*_p}\rangle}\frac{1}{k-k_p}
</math></center>
and we define a corresponding mode shape as
<center><math>
\zeta_{k_p}(x,k_p) =\frac{\langle \mathbf{F}(k_p),\mathbf{u}^*_{k^*_p} \rangle}{\langle\mathbf{u}_{k_p},\mathbf{M}'(k)\mathbf{u}^*_{k^*_p}\rangle}U(x)
</math></center>.
where <math>U(x)</math>

== Matlab Code ==

{{fixed body 2d bem code}}

=== Additional code ===

This program requires
* {{free surface dispersion equation code}}
* {{boundary element code}}

[[Category:Boundary Element Method]]
[[Category:Eigenfunction Matching Method]]

Boundary Element Method for a Fixed Body in Finite Depth

2009-11-25T02:21:58Z

Cfitzgerald: /* Singularity Expansion Method */

{{complete pages}}

== Introduction ==

We show how to solve for the wave scattering by a rigid body in constant [[Finite Depth]] using
the [[:Category:Boundary Element Method|Boundary Element Method]]. The method can be modified
to account for a body which can move and this is described in
[[Boundary Element Method for a Floating Body in Finite Depth]]

== Equations ==

{{boundary value problem for a fixed body}}

{{incident plane wave}}

{{sommerfeld radiation condition two dimensions}}

== Solution Method ==

We divide the domain into three regions, <math>x<-l</math>,
<math>x>r</math>, and <math>-l<x<r</math> so that the body surface is entirely in the
finite region.

=== Solution in the finite region ===

We use the [[:Category:Boundary Element Method|Boundary Element Method]]
in the finite region.

===Solution in the Semi-infinite Domains===

We now solve Laplace's equation in the semi-infinite domains. First consider
the domain on the left so that <math>\Omega = \left\{ x<-l,\;-h\leq z\leq 0\right\} </math>
The equations are
{{standard linear wave scattering equations without body condition}}
We have the following explicit boundary conditions
<center><math>
\phi =f\left( z\right) ,\;\;x=-\,l
</math></center>
and
<center><math>
\lim\limits_{x\rightarrow -\infty }\phi \left( x,z\right) = \phi_0
e^{-k_{0}x}
+R\phi_0
e^{k_{0}x},
</math></center>
where <math>f\left( z\right) </math> is
an arbitrary continuous function<math>.</math> Our aim is to find the outward normal
derivative of the potential on <math>x=-l</math> as a function of <math>\tilde{\phi}\left(
z\right) </math>. <math>\phi_0</math> and <math>k_0</math> will be defined shortly
when we separate variables, but are equivalent to the [[Sommerfeld Radiation Condition]].

Since the water depth is constant in these regions we can solve Laplace's
equation by separation of variables.

{{separation of variables for a free surface}}

=== Expansion of the potential ===

This gives us the following expression for the potential in the region <math>
\Omega </math>
<center><math>
\phi \left( x,z\right) = \phi_0(z)e^{-k_0 x}
+\sum_{m=0}^{\infty } a_m \phi _{m}\left( z\right)
e^{k_{m}\left( x+l\right) }
</math></center>
where <math>a_m</math> can be found from <math>\left. \phi \right| _{x=-l} = f(z)</math>
from the orthogonality of the vertical eigenfunctions. Therefore
<center><math>
a_m +\delta_{m0}e^{k_0 l} = \frac{1}{A_m} \int_{-h}^{0} f(z) \phi_m(z) \mathrm{d}z
</math></center>

The normal derivative of potential (with the normal outward from the interior region) we obtain
<center><math>
\partial_n\left. \phi \right| _{x=-l}
= -\partial_x \left. \phi \right| _{x=-l}
= k_0 \phi_0(z)e^{k_0 l}
- \sum_{m=0}^{\infty } k_m a_m \phi _{m}\left( z\right)
</math></center>

If we define <math>\mathbf{Q}</math> as
<center><math>
\mathbf{Q} \left[f\left( z\right) \right]=
-\sum_{m=0}^{\infty } \frac{k_{m}}{A_m} \left\{ \int_{-h}^{0} f\left( s\right) \phi
_{m}\left( s\right) \mathrm{d}s \right\} \phi _{m}\left( z\right)
</math></center>
we can write
<center><math>
\partial_n \left. \phi \right| _{x=-l} =\mathbf{Q} \left[ f(z) \right] + 2k_0 \phi_0e^{k_0l},
</math></center>

Similarly, we now consider the potential in the region
<math>\Omega = \left\{ x>r,\;-h\leq z\leq 0\right\} </math>
which satisfies exactly the equations as before except the boundary
condition is
<center><math>
\lim\limits_{x\rightarrow \infty }\phi \left( x,z\right) = \phi_0
T e^{-k_{0}x}
</math></center>
We solve again by separation of variables and obtain
<center><math>
\left. \partial_n \phi \right| _{x=r}=
\left. \partial_x \phi \right| _{x=r}=\mathbf{Q}\left[f\left( z\right)\right] ,
</math></center>
where the outward normal is with respect to the inner domain as before. Note that, if
the depth is different on each side then the matrix <math>\mathbf{Q}</math> will be different
(since we need to use a different depth solving the [[Dispersion Relation for a Free Surface]] etc.)

== Boundary Element Method ==

We have reduced the problem to Laplace's equation in a finite domain subject
to certain boundary conditions. These boundary
conditions give the outward normal derivative of the potential as a function
of the potential but this is not always a point-wise condition; on some
boundaries it is given by an integral equation. We must solve both Laplace's
equation and the integral equations numerically. We will solve Laplace's
equation by the [[:Category:Boundary Element Method|Boundary Element Method]] and the integral equations by
numerical integration. However, the same discretisation of the boundary will
be used for both numerical solutions.
We solve Laplace's equation by a modified constant panel method
which reduces it to the following matrix equation
<center><math>
\frac{1}{2}\vec{\phi}=\partial_n\mathbf{G}\vec{\phi}-\mathbf{G}\partial_n\vec{\phi}.
</math></center>
where <math>\vec{\phi}\mathcal{\ }</math>and <math>\partial_n\vec{\phi}</math>
are vectors which approximate the potential and its normal derivative
around the boundary <math>\partial \Omega </math>, and <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>
are matrices corresponding to the Green function and the outward normal
derivative of the Green function respectively. The method used to calculate
the elements of the matrices <math>\mathbf{G}</math> and <math>\partial_n\mathbf{G}</math> can be
found in [[:Category:Boundary Element Method|Boundary Element Method]].

The outward normal derivative of the potential, <math>\partial_n\vec{\phi},</math> and the
potential, <math>\vec{\phi},</math> are related by the conditions on the boundary <math>
\partial \Omega </math>. This can be expressed as
<center><math>
\partial_n\vec{\phi}=\mathbf{A}\,\vec{\phi}-\vec{f},
</math></center>
where <math>\mathbf{A}</math> is the block diagonal matrix given by
<center><math>
\mathbf{A}\mathbb{=}\left[
\begin{matrix}
\mathbf{Q} & & & \\
& \mathbf{Q} & & \\
& & \alpha \,\mathbf{I} & \\
& & & 0
\end{matrix}
\right] ,
</math></center>
where we have separated the boundaries to the semi-infinite domains, the free surface and the
sea floor and body boundary.
<math>\mathbf{Q}</math> is a matrix
approximation of the integral operator of the same name and <math>\vec{f}</math> is
the vector
<center><math>
\vec{f}=\left[
\begin{matrix}
-2k_{0}\phi_0 \,e^{k_{0}l } \\
0 \\
\vdots \\
0
\end{matrix}
\right] .
</math></center>

We obtain the following matrix equation for the
potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
which can be solved straightforwardly. The reflection and transmission
coefficients are calculated from the solution.

===Numerical Calculation of <math>\mathbf{Q}</math> ===

We begin by truncating to a finite number (<math>N</math>) of
evanescent modes,
<center><math>
\mathbf{Q}\left[ f \right] =-\sum_{m=0}^{N}k_{m}\int_{-h}^{0}
f\left( s\right) \phi_{m}\left( s\right) \mathrm{d}s
\frac{\phi _{m} \left( z\right)}{A_m} .
</math></center>
We calculate the integral
with the same panels as we used to approximate the integrals of the Green
function and its normal derivative . Similarly, we
assume that <math>f(s) \,</math> is constant over each panel and integrate <math>\phi
_{m}\left( s\right) </math> exactly. This gives us the
following matrix factorisation of <math>\mathbf{Q}</math>
<center><math>
\mathbf{Q}[f]=\mathbf{S}\,\mathbf{R}\,[f].
</math></center>
The components of the matrices <math>\mathbf{S}</math> and <math>\mathbf{R}</math> are
<center><math>
s_{jm} = -k_m\phi _{m}\left( z_{j}\right) ,
</math></center>
<center><math>
r_{mj} = \frac{1}{A_m} \int_{z_{j}-\Delta x/2}^{z_{j}+\Delta x/2}\phi
_{m}\left( s\right) \mathrm{d}s
</math></center>

=== Reflection and Transmission Coefficients ===

If we multiply the potential at the left (after subtracting the incident wave) and the right by
<math>\mathbf{R}</math> we can calculate the coefficients in the eigenfunction expansion, and
hence determine the reflection and transmission coefficient.
where <math>z_{j}</math> is the value of the <math>z</math> coordinate in the centre of the panel
and <math>\Delta x</math> is the panel length.

=== Singularity Expansion Method ===
The matrix equation for the potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
can be rewritten as
<center><math>
\mathbf{M}(k) \mathbf{x}=\mathbf{F}(k)
</math></center>
where <math>\mathbf{x}</math> represents the unknown potential values in the new notation. Resonant frequencies (or scattering frequencies) correspond to zero eigenvalues of the matrix <math>\mathbf{M}(k)</math> for a set of discrete points <math>k=k_p</math> in the upper half of the complex plane (due to the time dependence). Since
<center><math>
\mathbf{x}=\left( \mathbf{M}(k)\right)^{-1}\mathbf{F}(k)
</math></center>
then <math>\mathbf{x}</math> will be singular for these scattering frequencies. Each scattering frequency, corresponding to a zero eigenvalue of <math>\mathbf{M}(k)</math> will possess a corresponding eigenvector <math>\mathbf{u}_{k_p}</math> satisfying
<center><math>
\mathbf{M}(k_p)\mathbf{u}_{k_p}=0
</math></center>
and also a related conjugate transpose eigenvector <math>\mathbf{u}^{*}_{k_p}</math> satisfying
<center><math>
\mathbf{M}(k_p)^{A}\mathbf{u}^*_{k_p}=0.
</math></center>
In the vicinity of a scattering frequency it can be shown that the vector of unknowns satisfies
<center><math>
\mathbf{x}(k)\approx \frac{\langle \mathbf{F}(k_p),\mathbf{u}^*_{k^*_p} \rangle}{\langle\mathbf{u}_{k_p},\mathbf{M}'(k_p)\mathbf{u}^*_{k^*_p}\rangle}\frac{1}{k-k_p}
</math></center>
and we define a corresponding mode shape as
<center><math>
\zeta_{k_p}(x,k_p) =\frac{\langle \mathbf{F}(k_p),\mathbf{u}^* \rangle}{\langle\mathbf{u},\mathbf{M}'(k)\mathbf{u}^*\rangle}U(x)
</math></center>.
where <math>U(x)</math>

== Matlab Code ==

{{fixed body 2d bem code}}

=== Additional code ===

This program requires
* {{free surface dispersion equation code}}
* {{boundary element code}}

[[Category:Boundary Element Method]]
[[Category:Eigenfunction Matching Method]]

Boundary Element Method for a Fixed Body in Finite Depth

2009-11-25T02:10:41Z

Cfitzgerald: /* Singularity Expansion Method */

{{complete pages}}

== Introduction ==

We show how to solve for the wave scattering by a rigid body in constant [[Finite Depth]] using
the [[:Category:Boundary Element Method|Boundary Element Method]]. The method can be modified
to account for a body which can move and this is described in
[[Boundary Element Method for a Floating Body in Finite Depth]]

== Equations ==

{{boundary value problem for a fixed body}}

{{incident plane wave}}

{{sommerfeld radiation condition two dimensions}}

== Solution Method ==

We divide the domain into three regions, <math>x<-l</math>,
<math>x>r</math>, and <math>-l<x<r</math> so that the body surface is entirely in the
finite region.

=== Solution in the finite region ===

We use the [[:Category:Boundary Element Method|Boundary Element Method]]
in the finite region.

===Solution in the Semi-infinite Domains===

We now solve Laplace's equation in the semi-infinite domains. First consider
the domain on the left so that <math>\Omega = \left\{ x<-l,\;-h\leq z\leq 0\right\} </math>
The equations are
{{standard linear wave scattering equations without body condition}}
We have the following explicit boundary conditions
<center><math>
\phi =f\left( z\right) ,\;\;x=-\,l
</math></center>
and
<center><math>
\lim\limits_{x\rightarrow -\infty }\phi \left( x,z\right) = \phi_0
e^{-k_{0}x}
+R\phi_0
e^{k_{0}x},
</math></center>
where <math>f\left( z\right) </math> is
an arbitrary continuous function<math>.</math> Our aim is to find the outward normal
derivative of the potential on <math>x=-l</math> as a function of <math>\tilde{\phi}\left(
z\right) </math>. <math>\phi_0</math> and <math>k_0</math> will be defined shortly
when we separate variables, but are equivalent to the [[Sommerfeld Radiation Condition]].

Since the water depth is constant in these regions we can solve Laplace's
equation by separation of variables.

{{separation of variables for a free surface}}

=== Expansion of the potential ===

This gives us the following expression for the potential in the region <math>
\Omega </math>
<center><math>
\phi \left( x,z\right) = \phi_0(z)e^{-k_0 x}
+\sum_{m=0}^{\infty } a_m \phi _{m}\left( z\right)
e^{k_{m}\left( x+l\right) }
</math></center>
where <math>a_m</math> can be found from <math>\left. \phi \right| _{x=-l} = f(z)</math>
from the orthogonality of the vertical eigenfunctions. Therefore
<center><math>
a_m +\delta_{m0}e^{k_0 l} = \frac{1}{A_m} \int_{-h}^{0} f(z) \phi_m(z) \mathrm{d}z
</math></center>

The normal derivative of potential (with the normal outward from the interior region) we obtain
<center><math>
\partial_n\left. \phi \right| _{x=-l}
= -\partial_x \left. \phi \right| _{x=-l}
= k_0 \phi_0(z)e^{k_0 l}
- \sum_{m=0}^{\infty } k_m a_m \phi _{m}\left( z\right)
</math></center>

If we define <math>\mathbf{Q}</math> as
<center><math>
\mathbf{Q} \left[f\left( z\right) \right]=
-\sum_{m=0}^{\infty } \frac{k_{m}}{A_m} \left\{ \int_{-h}^{0} f\left( s\right) \phi
_{m}\left( s\right) \mathrm{d}s \right\} \phi _{m}\left( z\right)
</math></center>
we can write
<center><math>
\partial_n \left. \phi \right| _{x=-l} =\mathbf{Q} \left[ f(z) \right] + 2k_0 \phi_0e^{k_0l},
</math></center>

Similarly, we now consider the potential in the region
<math>\Omega = \left\{ x>r,\;-h\leq z\leq 0\right\} </math>
which satisfies exactly the equations as before except the boundary
condition is
<center><math>
\lim\limits_{x\rightarrow \infty }\phi \left( x,z\right) = \phi_0
T e^{-k_{0}x}
</math></center>
We solve again by separation of variables and obtain
<center><math>
\left. \partial_n \phi \right| _{x=r}=
\left. \partial_x \phi \right| _{x=r}=\mathbf{Q}\left[f\left( z\right)\right] ,
</math></center>
where the outward normal is with respect to the inner domain as before. Note that, if
the depth is different on each side then the matrix <math>\mathbf{Q}</math> will be different
(since we need to use a different depth solving the [[Dispersion Relation for a Free Surface]] etc.)

== Boundary Element Method ==

We have reduced the problem to Laplace's equation in a finite domain subject
to certain boundary conditions. These boundary
conditions give the outward normal derivative of the potential as a function
of the potential but this is not always a point-wise condition; on some
boundaries it is given by an integral equation. We must solve both Laplace's
equation and the integral equations numerically. We will solve Laplace's
equation by the [[:Category:Boundary Element Method|Boundary Element Method]] and the integral equations by
numerical integration. However, the same discretisation of the boundary will
be used for both numerical solutions.
We solve Laplace's equation by a modified constant panel method
which reduces it to the following matrix equation
<center><math>
\frac{1}{2}\vec{\phi}=\partial_n\mathbf{G}\vec{\phi}-\mathbf{G}\partial_n\vec{\phi}.
</math></center>
where <math>\vec{\phi}\mathcal{\ }</math>and <math>\partial_n\vec{\phi}</math>
are vectors which approximate the potential and its normal derivative
around the boundary <math>\partial \Omega </math>, and <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>
are matrices corresponding to the Green function and the outward normal
derivative of the Green function respectively. The method used to calculate
the elements of the matrices <math>\mathbf{G}</math> and <math>\partial_n\mathbf{G}</math> can be
found in [[:Category:Boundary Element Method|Boundary Element Method]].

The outward normal derivative of the potential, <math>\partial_n\vec{\phi},</math> and the
potential, <math>\vec{\phi},</math> are related by the conditions on the boundary <math>
\partial \Omega </math>. This can be expressed as
<center><math>
\partial_n\vec{\phi}=\mathbf{A}\,\vec{\phi}-\vec{f},
</math></center>
where <math>\mathbf{A}</math> is the block diagonal matrix given by
<center><math>
\mathbf{A}\mathbb{=}\left[
\begin{matrix}
\mathbf{Q} & & & \\
& \mathbf{Q} & & \\
& & \alpha \,\mathbf{I} & \\
& & & 0
\end{matrix}
\right] ,
</math></center>
where we have separated the boundaries to the semi-infinite domains, the free surface and the
sea floor and body boundary.
<math>\mathbf{Q}</math> is a matrix
approximation of the integral operator of the same name and <math>\vec{f}</math> is
the vector
<center><math>
\vec{f}=\left[
\begin{matrix}
-2k_{0}\phi_0 \,e^{k_{0}l } \\
0 \\
\vdots \\
0
\end{matrix}
\right] .
</math></center>

We obtain the following matrix equation for the
potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
which can be solved straightforwardly. The reflection and transmission
coefficients are calculated from the solution.

===Numerical Calculation of <math>\mathbf{Q}</math> ===

We begin by truncating to a finite number (<math>N</math>) of
evanescent modes,
<center><math>
\mathbf{Q}\left[ f \right] =-\sum_{m=0}^{N}k_{m}\int_{-h}^{0}
f\left( s\right) \phi_{m}\left( s\right) \mathrm{d}s
\frac{\phi _{m} \left( z\right)}{A_m} .
</math></center>
We calculate the integral
with the same panels as we used to approximate the integrals of the Green
function and its normal derivative . Similarly, we
assume that <math>f(s) \,</math> is constant over each panel and integrate <math>\phi
_{m}\left( s\right) </math> exactly. This gives us the
following matrix factorisation of <math>\mathbf{Q}</math>
<center><math>
\mathbf{Q}[f]=\mathbf{S}\,\mathbf{R}\,[f].
</math></center>
The components of the matrices <math>\mathbf{S}</math> and <math>\mathbf{R}</math> are
<center><math>
s_{jm} = -k_m\phi _{m}\left( z_{j}\right) ,
</math></center>
<center><math>
r_{mj} = \frac{1}{A_m} \int_{z_{j}-\Delta x/2}^{z_{j}+\Delta x/2}\phi
_{m}\left( s\right) \mathrm{d}s
</math></center>

=== Reflection and Transmission Coefficients ===

If we multiply the potential at the left (after subtracting the incident wave) and the right by
<math>\mathbf{R}</math> we can calculate the coefficients in the eigenfunction expansion, and
hence determine the reflection and transmission coefficient.
where <math>z_{j}</math> is the value of the <math>z</math> coordinate in the centre of the panel
and <math>\Delta x</math> is the panel length.

=== Singularity Expansion Method ===
The matrix equation for the potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
can be rewritten as
<center><math>
\mathbf{M}(k) \mathbf{x}=\mathbf{F}(k)
</math></center>
where <math>\mathbf{x}</math> represents the unknown potential values in the new notation. Resonant frequencies (or scattering frequencies) correspond to zero eigenvalues of the matrix <math>\mathbf{M}(k)</math> for a set of discrete points <math>k=k_p</math> in the upper half of the complex plane (due to the time dependence). Since
<center><math>
\mathbf{x}=\left( \mathbf{M}(k)\right)^{-1}\mathbf{F}(k)
</math></center>
then <math>\mathbf{x}</math> will be singular for these scattering frequencies. Each scattering frequency, corresponding to a zero eigenvalue of <math>\mathbf{M}(k)</math> will possess a corresponding eigenvector <math>\mathbf{u}_{k_p}</math> satisfying
<center><math>
\mathbf{M}(k_p)\mathbf{u}_{k_p}=0
</math></center>
and also a related conjugate transpose eigenvector <math>\mathbf{u}^{*}_{k_p}</math> satisfying
<center><math>
\mathbf{M}(k_p)^{A}\mathbf{u}^*_{k_p}=0.
</math></center>
In the vicinity of a scattering frequency it can be shown that the vector of unknowns satisfies
<center><math>
\mathbf{x}(k)\approx \frac{\langle \mathbf{F}(k_p),\mathbf{u}^* \rangle}{\langle\mathbf{u},\mathbf{M}'(k_p)\mathbf{u}^*\rangle}\frac{1}{k-k_p}
</math></center>
and we define a corresponding mode shape as
<center><math>
\zeta_{k_p}(x,k_p) =\frac{\langle \mathbf{F}(k_p),\mathbf{u}^* \rangle}{\langle\mathbf{u},\mathbf{M}'(k)\mathbf{u}^*\rangle}U(x)
</math></center>.
where <math>U(x)</math>

== Matlab Code ==

{{fixed body 2d bem code}}

=== Additional code ===

This program requires
* {{free surface dispersion equation code}}
* {{boundary element code}}

[[Category:Boundary Element Method]]
[[Category:Eigenfunction Matching Method]]

Boundary Element Method for a Fixed Body in Finite Depth

2009-11-24T22:53:54Z

Cfitzgerald: /* Singularity Expansion Method */

{{complete pages}}

== Introduction ==

We show how to solve for the wave scattering by a rigid body in constant [[Finite Depth]] using
the [[:Category:Boundary Element Method|Boundary Element Method]]. The method can be modified
to account for a body which can move and this is described in
[[Boundary Element Method for a Floating Body in Finite Depth]]

== Equations ==

{{boundary value problem for a fixed body}}

{{incident plane wave}}

{{sommerfeld radiation condition two dimensions}}

== Solution Method ==

We divide the domain into three regions, <math>x<-l</math>,
<math>x>r</math>, and <math>-l<x<r</math> so that the body surface is entirely in the
finite region.

=== Solution in the finite region ===

We use the [[:Category:Boundary Element Method|Boundary Element Method]]
in the finite region.

===Solution in the Semi-infinite Domains===

We now solve Laplace's equation in the semi-infinite domains. First consider
the domain on the left so that <math>\Omega = \left\{ x<-l,\;-h\leq z\leq 0\right\} </math>
The equations are
{{standard linear wave scattering equations without body condition}}
We have the following explicit boundary conditions
<center><math>
\phi =f\left( z\right) ,\;\;x=-\,l
</math></center>
and
<center><math>
\lim\limits_{x\rightarrow -\infty }\phi \left( x,z\right) = \phi_0
e^{-k_{0}x}
+R\phi_0
e^{k_{0}x},
</math></center>
where <math>f\left( z\right) </math> is
an arbitrary continuous function<math>.</math> Our aim is to find the outward normal
derivative of the potential on <math>x=-l</math> as a function of <math>\tilde{\phi}\left(
z\right) </math>. <math>\phi_0</math> and <math>k_0</math> will be defined shortly
when we separate variables, but are equivalent to the [[Sommerfeld Radiation Condition]].

Since the water depth is constant in these regions we can solve Laplace's
equation by separation of variables.

{{separation of variables for a free surface}}

=== Expansion of the potential ===

This gives us the following expression for the potential in the region <math>
\Omega </math>
<center><math>
\phi \left( x,z\right) = \phi_0(z)e^{-k_0 x}
+\sum_{m=0}^{\infty } a_m \phi _{m}\left( z\right)
e^{k_{m}\left( x+l\right) }
</math></center>
where <math>a_m</math> can be found from <math>\left. \phi \right| _{x=-l} = f(z)</math>
from the orthogonality of the vertical eigenfunctions. Therefore
<center><math>
a_m +\delta_{m0}e^{k_0 l} = \frac{1}{A_m} \int_{-h}^{0} f(z) \phi_m(z) \mathrm{d}z
</math></center>

The normal derivative of potential (with the normal outward from the interior region) we obtain
<center><math>
\partial_n\left. \phi \right| _{x=-l}
= -\partial_x \left. \phi \right| _{x=-l}
= k_0 \phi_0(z)e^{k_0 l}
- \sum_{m=0}^{\infty } k_m a_m \phi _{m}\left( z\right)
</math></center>

If we define <math>\mathbf{Q}</math> as
<center><math>
\mathbf{Q} \left[f\left( z\right) \right]=
-\sum_{m=0}^{\infty } \frac{k_{m}}{A_m} \left\{ \int_{-h}^{0} f\left( s\right) \phi
_{m}\left( s\right) \mathrm{d}s \right\} \phi _{m}\left( z\right)
</math></center>
we can write
<center><math>
\partial_n \left. \phi \right| _{x=-l} =\mathbf{Q} \left[ f(z) \right] + 2k_0 \phi_0e^{k_0l},
</math></center>

Similarly, we now consider the potential in the region
<math>\Omega = \left\{ x>r,\;-h\leq z\leq 0\right\} </math>
which satisfies exactly the equations as before except the boundary
condition is
<center><math>
\lim\limits_{x\rightarrow \infty }\phi \left( x,z\right) = \phi_0
T e^{-k_{0}x}
</math></center>
We solve again by separation of variables and obtain
<center><math>
\left. \partial_n \phi \right| _{x=r}=
\left. \partial_x \phi \right| _{x=r}=\mathbf{Q}\left[f\left( z\right)\right] ,
</math></center>
where the outward normal is with respect to the inner domain as before. Note that, if
the depth is different on each side then the matrix <math>\mathbf{Q}</math> will be different
(since we need to use a different depth solving the [[Dispersion Relation for a Free Surface]] etc.)

== Boundary Element Method ==

We have reduced the problem to Laplace's equation in a finite domain subject
to certain boundary conditions. These boundary
conditions give the outward normal derivative of the potential as a function
of the potential but this is not always a point-wise condition; on some
boundaries it is given by an integral equation. We must solve both Laplace's
equation and the integral equations numerically. We will solve Laplace's
equation by the [[:Category:Boundary Element Method|Boundary Element Method]] and the integral equations by
numerical integration. However, the same discretisation of the boundary will
be used for both numerical solutions.
We solve Laplace's equation by a modified constant panel method
which reduces it to the following matrix equation
<center><math>
\frac{1}{2}\vec{\phi}=\partial_n\mathbf{G}\vec{\phi}-\mathbf{G}\partial_n\vec{\phi}.
</math></center>
where <math>\vec{\phi}\mathcal{\ }</math>and <math>\partial_n\vec{\phi}</math>
are vectors which approximate the potential and its normal derivative
around the boundary <math>\partial \Omega </math>, and <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>
are matrices corresponding to the Green function and the outward normal
derivative of the Green function respectively. The method used to calculate
the elements of the matrices <math>\mathbf{G}</math> and <math>\partial_n\mathbf{G}</math> can be
found in [[:Category:Boundary Element Method|Boundary Element Method]].

The outward normal derivative of the potential, <math>\partial_n\vec{\phi},</math> and the
potential, <math>\vec{\phi},</math> are related by the conditions on the boundary <math>
\partial \Omega </math>. This can be expressed as
<center><math>
\partial_n\vec{\phi}=\mathbf{A}\,\vec{\phi}-\vec{f},
</math></center>
where <math>\mathbf{A}</math> is the block diagonal matrix given by
<center><math>
\mathbf{A}\mathbb{=}\left[
\begin{matrix}
\mathbf{Q} & & & \\
& \mathbf{Q} & & \\
& & \alpha \,\mathbf{I} & \\
& & & 0
\end{matrix}
\right] ,
</math></center>
where we have separated the boundaries to the semi-infinite domains, the free surface and the
sea floor and body boundary.
<math>\mathbf{Q}</math> is a matrix
approximation of the integral operator of the same name and <math>\vec{f}</math> is
the vector
<center><math>
\vec{f}=\left[
\begin{matrix}
-2k_{0}\phi_0 \,e^{k_{0}l } \\
0 \\
\vdots \\
0
\end{matrix}
\right] .
</math></center>

We obtain the following matrix equation for the
potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
which can be solved straightforwardly. The reflection and transmission
coefficients are calculated from the solution.

===Numerical Calculation of <math>\mathbf{Q}</math> ===

We begin by truncating to a finite number (<math>N</math>) of
evanescent modes,
<center><math>
\mathbf{Q}\left[ f \right] =-\sum_{m=0}^{N}k_{m}\int_{-h}^{0}
f\left( s\right) \phi_{m}\left( s\right) \mathrm{d}s
\frac{\phi _{m} \left( z\right)}{A_m} .
</math></center>
We calculate the integral
with the same panels as we used to approximate the integrals of the Green
function and its normal derivative . Similarly, we
assume that <math>f(s) \,</math> is constant over each panel and integrate <math>\phi
_{m}\left( s\right) </math> exactly. This gives us the
following matrix factorisation of <math>\mathbf{Q}</math>
<center><math>
\mathbf{Q}[f]=\mathbf{S}\,\mathbf{R}\,[f].
</math></center>
The components of the matrices <math>\mathbf{S}</math> and <math>\mathbf{R}</math> are
<center><math>
s_{jm} = -k_m\phi _{m}\left( z_{j}\right) ,
</math></center>
<center><math>
r_{mj} = \frac{1}{A_m} \int_{z_{j}-\Delta x/2}^{z_{j}+\Delta x/2}\phi
_{m}\left( s\right) \mathrm{d}s
</math></center>

=== Reflection and Transmission Coefficients ===

If we multiply the potential at the left (after subtracting the incident wave) and the right by
<math>\mathbf{R}</math> we can calculate the coefficients in the eigenfunction expansion, and
hence determine the reflection and transmission coefficient.
where <math>z_{j}</math> is the value of the <math>z</math> coordinate in the centre of the panel
and <math>\Delta x</math> is the panel length.

=== Singularity Expansion Method ===
The matrix equation for the potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
can be rewritten as
<center><math>
\mathbf{M}(k) \mathbf{x}=\mathbf{F}(k)
</math></center>
where <math>\mathbf{x}</math> represents the unknown potential values in the new notation. Resonant frequencies (or scattering frequencies) correspond to zero eigenvalues of the matrix <math>\mathbf{M}(k)</math> for a set of discrete points <math>k=k_p</math> in the upper half of the complex plane (due to the time dependence). Since
<center><math>
\mathbf{x}=\left( \mathbf{M}(k)\right)^{-1}\mathbf{F}(k)
</math></center>
then <math>\mathbf{x}</math> will be singular for these scattering frequencies. Each scattering frequency, corresponding to a zero eigenvalue of <math>\mathbf{M}(k)</math> will possess a corresponding eigenvector <math>\mathbf{u}_{k_p}</math> satisfying
<center><math>
\mathbf{M}(k_p)\mathbf{u}_{k_p}=0
</math></center>
and also a related conjugate transpose eigenvector <math>\mathbf{u}^{*}_{k_p}</math> satisfying
<center><math>
\mathbf{M}(k_p)^{A}\mathbf{u}^*_{k_p}=0.
</math></center>
In the vicinity of a scattering frequency it can be shown that the vector of unknowns satisfies
<center><math>
\mathbf{x}(k)\approx \frac{\langle \mathbf{F}(k),\mathbf{u}^* \rangle}{\langle\mathbf{u},\mathbf{M}'(k)\mathbf{u}^*\rangle}\frac{1}{k-k_p}
</math></center>
and we define a corresponding mode shape as
<center><math>
\zeta_{k_p}(x,k_p) =\frac{\langle \mathbf{F}(k_p),\mathbf{u}^* \rangle}{\langle\mathbf{u},\mathbf{M}'(k)\mathbf{u}^*\rangle}U(x)
</math></center>.
where <math>U(x)</math>

== Matlab Code ==

{{fixed body 2d bem code}}

=== Additional code ===

This program requires
* {{free surface dispersion equation code}}
* {{boundary element code}}

[[Category:Boundary Element Method]]
[[Category:Eigenfunction Matching Method]]

Boundary Element Method for a Fixed Body in Finite Depth

2009-11-24T22:42:21Z

Cfitzgerald: /* Singularity Expansion Method */

{{complete pages}}

== Introduction ==

We show how to solve for the wave scattering by a rigid body in constant [[Finite Depth]] using
the [[:Category:Boundary Element Method|Boundary Element Method]]. The method can be modified
to account for a body which can move and this is described in
[[Boundary Element Method for a Floating Body in Finite Depth]]

== Equations ==

{{boundary value problem for a fixed body}}

{{incident plane wave}}

{{sommerfeld radiation condition two dimensions}}

== Solution Method ==

We divide the domain into three regions, <math>x<-l</math>,
<math>x>r</math>, and <math>-l<x<r</math> so that the body surface is entirely in the
finite region.

=== Solution in the finite region ===

We use the [[:Category:Boundary Element Method|Boundary Element Method]]
in the finite region.

===Solution in the Semi-infinite Domains===

We now solve Laplace's equation in the semi-infinite domains. First consider
the domain on the left so that <math>\Omega = \left\{ x<-l,\;-h\leq z\leq 0\right\} </math>
The equations are
{{standard linear wave scattering equations without body condition}}
We have the following explicit boundary conditions
<center><math>
\phi =f\left( z\right) ,\;\;x=-\,l
</math></center>
and
<center><math>
\lim\limits_{x\rightarrow -\infty }\phi \left( x,z\right) = \phi_0
e^{-k_{0}x}
+R\phi_0
e^{k_{0}x},
</math></center>
where <math>f\left( z\right) </math> is
an arbitrary continuous function<math>.</math> Our aim is to find the outward normal
derivative of the potential on <math>x=-l</math> as a function of <math>\tilde{\phi}\left(
z\right) </math>. <math>\phi_0</math> and <math>k_0</math> will be defined shortly
when we separate variables, but are equivalent to the [[Sommerfeld Radiation Condition]].

Since the water depth is constant in these regions we can solve Laplace's
equation by separation of variables.

{{separation of variables for a free surface}}

=== Expansion of the potential ===

This gives us the following expression for the potential in the region <math>
\Omega </math>
<center><math>
\phi \left( x,z\right) = \phi_0(z)e^{-k_0 x}
+\sum_{m=0}^{\infty } a_m \phi _{m}\left( z\right)
e^{k_{m}\left( x+l\right) }
</math></center>
where <math>a_m</math> can be found from <math>\left. \phi \right| _{x=-l} = f(z)</math>
from the orthogonality of the vertical eigenfunctions. Therefore
<center><math>
a_m +\delta_{m0}e^{k_0 l} = \frac{1}{A_m} \int_{-h}^{0} f(z) \phi_m(z) \mathrm{d}z
</math></center>

The normal derivative of potential (with the normal outward from the interior region) we obtain
<center><math>
\partial_n\left. \phi \right| _{x=-l}
= -\partial_x \left. \phi \right| _{x=-l}
= k_0 \phi_0(z)e^{k_0 l}
- \sum_{m=0}^{\infty } k_m a_m \phi _{m}\left( z\right)
</math></center>

If we define <math>\mathbf{Q}</math> as
<center><math>
\mathbf{Q} \left[f\left( z\right) \right]=
-\sum_{m=0}^{\infty } \frac{k_{m}}{A_m} \left\{ \int_{-h}^{0} f\left( s\right) \phi
_{m}\left( s\right) \mathrm{d}s \right\} \phi _{m}\left( z\right)
</math></center>
we can write
<center><math>
\partial_n \left. \phi \right| _{x=-l} =\mathbf{Q} \left[ f(z) \right] + 2k_0 \phi_0e^{k_0l},
</math></center>

Similarly, we now consider the potential in the region
<math>\Omega = \left\{ x>r,\;-h\leq z\leq 0\right\} </math>
which satisfies exactly the equations as before except the boundary
condition is
<center><math>
\lim\limits_{x\rightarrow \infty }\phi \left( x,z\right) = \phi_0
T e^{-k_{0}x}
</math></center>
We solve again by separation of variables and obtain
<center><math>
\left. \partial_n \phi \right| _{x=r}=
\left. \partial_x \phi \right| _{x=r}=\mathbf{Q}\left[f\left( z\right)\right] ,
</math></center>
where the outward normal is with respect to the inner domain as before. Note that, if
the depth is different on each side then the matrix <math>\mathbf{Q}</math> will be different
(since we need to use a different depth solving the [[Dispersion Relation for a Free Surface]] etc.)

== Boundary Element Method ==

We have reduced the problem to Laplace's equation in a finite domain subject
to certain boundary conditions. These boundary
conditions give the outward normal derivative of the potential as a function
of the potential but this is not always a point-wise condition; on some
boundaries it is given by an integral equation. We must solve both Laplace's
equation and the integral equations numerically. We will solve Laplace's
equation by the [[:Category:Boundary Element Method|Boundary Element Method]] and the integral equations by
numerical integration. However, the same discretisation of the boundary will
be used for both numerical solutions.
We solve Laplace's equation by a modified constant panel method
which reduces it to the following matrix equation
<center><math>
\frac{1}{2}\vec{\phi}=\partial_n\mathbf{G}\vec{\phi}-\mathbf{G}\partial_n\vec{\phi}.
</math></center>
where <math>\vec{\phi}\mathcal{\ }</math>and <math>\partial_n\vec{\phi}</math>
are vectors which approximate the potential and its normal derivative
around the boundary <math>\partial \Omega </math>, and <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>
are matrices corresponding to the Green function and the outward normal
derivative of the Green function respectively. The method used to calculate
the elements of the matrices <math>\mathbf{G}</math> and <math>\partial_n\mathbf{G}</math> can be
found in [[:Category:Boundary Element Method|Boundary Element Method]].

The outward normal derivative of the potential, <math>\partial_n\vec{\phi},</math> and the
potential, <math>\vec{\phi},</math> are related by the conditions on the boundary <math>
\partial \Omega </math>. This can be expressed as
<center><math>
\partial_n\vec{\phi}=\mathbf{A}\,\vec{\phi}-\vec{f},
</math></center>
where <math>\mathbf{A}</math> is the block diagonal matrix given by
<center><math>
\mathbf{A}\mathbb{=}\left[
\begin{matrix}
\mathbf{Q} & & & \\
& \mathbf{Q} & & \\
& & \alpha \,\mathbf{I} & \\
& & & 0
\end{matrix}
\right] ,
</math></center>
where we have separated the boundaries to the semi-infinite domains, the free surface and the
sea floor and body boundary.
<math>\mathbf{Q}</math> is a matrix
approximation of the integral operator of the same name and <math>\vec{f}</math> is
the vector
<center><math>
\vec{f}=\left[
\begin{matrix}
-2k_{0}\phi_0 \,e^{k_{0}l } \\
0 \\
\vdots \\
0
\end{matrix}
\right] .
</math></center>

We obtain the following matrix equation for the
potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
which can be solved straightforwardly. The reflection and transmission
coefficients are calculated from the solution.

===Numerical Calculation of <math>\mathbf{Q}</math> ===

We begin by truncating to a finite number (<math>N</math>) of
evanescent modes,
<center><math>
\mathbf{Q}\left[ f \right] =-\sum_{m=0}^{N}k_{m}\int_{-h}^{0}
f\left( s\right) \phi_{m}\left( s\right) \mathrm{d}s
\frac{\phi _{m} \left( z\right)}{A_m} .
</math></center>
We calculate the integral
with the same panels as we used to approximate the integrals of the Green
function and its normal derivative . Similarly, we
assume that <math>f(s) \,</math> is constant over each panel and integrate <math>\phi
_{m}\left( s\right) </math> exactly. This gives us the
following matrix factorisation of <math>\mathbf{Q}</math>
<center><math>
\mathbf{Q}[f]=\mathbf{S}\,\mathbf{R}\,[f].
</math></center>
The components of the matrices <math>\mathbf{S}</math> and <math>\mathbf{R}</math> are
<center><math>
s_{jm} = -k_m\phi _{m}\left( z_{j}\right) ,
</math></center>
<center><math>
r_{mj} = \frac{1}{A_m} \int_{z_{j}-\Delta x/2}^{z_{j}+\Delta x/2}\phi
_{m}\left( s\right) \mathrm{d}s
</math></center>

=== Reflection and Transmission Coefficients ===

If we multiply the potential at the left (after subtracting the incident wave) and the right by
<math>\mathbf{R}</math> we can calculate the coefficients in the eigenfunction expansion, and
hence determine the reflection and transmission coefficient.
where <math>z_{j}</math> is the value of the <math>z</math> coordinate in the centre of the panel
and <math>\Delta x</math> is the panel length.

=== Singularity Expansion Method ===
The matrix equation for the potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
can be rewritten as
<center><math>
\mathbf{M}(k) \mathbf{x}=\mathbf{F}(k)
</math></center>
where <math>\mathbf{x}</math> represents the unknown potential values in the new notation. Resonant frequencies (or scattering frequencies) correspond to zero eigenvalues of the matrix <math>\mathbf{M}(k)</math> for a set of discrete points <math>k=k_p</math> in the upper half of the complex plane (due to the time dependence). Since
<center><math>
\mathbf{x}=\left( \mathbf{M}(k)\right)^{-1}\mathbf{F}(k)
</math></center>
then <math>\mathbf{x}</math> will be singular for these scattering frequencies. Each scattering frequency, corresponding to a zero eigenvalue of <math>\mathbf{M}(k)</math> will possess a corresponding eigenvector <math>\mathbf{u}_{k_p}</math> satisfying
<center><math>
\mathbf{M}(k_p)\mathbf{u}_{k_p}=0
</math></center>
and also a related conjugate transpose eigenvector <math>\mathbf{u}^{*}_{k_p}</math> satisfying
<center><math>
\mathbf{M}(k_p)^{A}\mathbf{u}^*_{k_p}=0.
</math></center>
In the vicinity of a scattering frequency it can be shown that the vector of unknowns satisfies
<center><math>
\mathbf{x}(k)\approx \frac{\langle \mathbf{F}(k),\mathbf{u}^* \rangle}{\langle\mathbf{u},\mathbf{M}'(k)\mathbf{u}^*\rangle}\frac{1}{k-k_p}
</math></center>
and we define a corresponding shape function as
<center><math>
\zeta_{k_p}(x,k_p) =\frac{\langle \mathbf{F}(k_p),\mathbf{u}^* \rangle}{\langle\mathbf{u},\mathbf{M}'(k)\mathbf{u}^*\rangle}\mathbf{x}
</math></center>.

== Matlab Code ==

{{fixed body 2d bem code}}

=== Additional code ===

This program requires
* {{free surface dispersion equation code}}
* {{boundary element code}}

[[Category:Boundary Element Method]]
[[Category:Eigenfunction Matching Method]]

Boundary Element Method for a Fixed Body in Finite Depth

2009-11-24T04:33:28Z

Cfitzgerald: /* Boundary Element Method */

{{complete pages}}

== Introduction ==

We show how to solve for the wave scattering by a rigid body in constant [[Finite Depth]] using
the [[:Category:Boundary Element Method|Boundary Element Method]]. The method can be modified
to account for a body which can move and this is described in
[[Boundary Element Method for a Floating Body in Finite Depth]]

== Equations ==

{{boundary value problem for a fixed body}}

{{incident plane wave}}

{{sommerfeld radiation condition two dimensions}}

== Solution Method ==

We divide the domain into three regions, <math>x<-l</math>,
<math>x>r</math>, and <math>-l<x<r</math> so that the body surface is entirely in the
finite region.

=== Solution in the finite region ===

We use the [[:Category:Boundary Element Method|Boundary Element Method]]
in the finite region.

===Solution in the Semi-infinite Domains===

We now solve Laplace's equation in the semi-infinite domains. First consider
the domain on the left so that <math>\Omega = \left\{ x<-l,\;-h\leq z\leq 0\right\} </math>
The equations are
{{standard linear wave scattering equations without body condition}}
We have the following explicit boundary conditions
<center><math>
\phi =f\left( z\right) ,\;\;x=-\,l
</math></center>
and
<center><math>
\lim\limits_{x\rightarrow -\infty }\phi \left( x,z\right) = \phi_0
e^{-k_{0}x}
+R\phi_0
e^{k_{0}x},
</math></center>
where <math>f\left( z\right) </math> is
an arbitrary continuous function<math>.</math> Our aim is to find the outward normal
derivative of the potential on <math>x=-l</math> as a function of <math>\tilde{\phi}\left(
z\right) </math>. <math>\phi_0</math> and <math>k_0</math> will be defined shortly
when we separate variables, but are equivalent to the [[Sommerfeld Radiation Condition]].

Since the water depth is constant in these regions we can solve Laplace's
equation by separation of variables.

{{separation of variables for a free surface}}

=== Expansion of the potential ===

This gives us the following expression for the potential in the region <math>
\Omega </math>
<center><math>
\phi \left( x,z\right) = \phi_0(z)e^{-k_0 x}
+\sum_{m=0}^{\infty } a_m \phi _{m}\left( z\right)
e^{k_{m}\left( x+l\right) }
</math></center>
where <math>a_m</math> can be found from <math>\left. \phi \right| _{x=-l} = f(z)</math>
from the orthogonality of the vertical eigenfunctions. Therefore
<center><math>
a_m +\delta_{m0}e^{k_0 l} = \frac{1}{A_m} \int_{-h}^{0} f(z) \phi_m(z) \mathrm{d}z
</math></center>

The normal derivative of potential (with the normal outward from the interior region) we obtain
<center><math>
\partial_n\left. \phi \right| _{x=-l}
= -\partial_x \left. \phi \right| _{x=-l}
= k_0 \phi_0(z)e^{k_0 l}
- \sum_{m=0}^{\infty } k_m a_m \phi _{m}\left( z\right)
</math></center>

If we define <math>\mathbf{Q}</math> as
<center><math>
\mathbf{Q} \left[f\left( z\right) \right]=
-\sum_{m=0}^{\infty } \frac{k_{m}}{A_m} \left\{ \int_{-h}^{0} f\left( s\right) \phi
_{m}\left( s\right) \mathrm{d}s \right\} \phi _{m}\left( z\right)
</math></center>
we can write
<center><math>
\partial_n \left. \phi \right| _{x=-l} =\mathbf{Q} \left[ f(z) \right] + 2k_0 \phi_0e^{k_0l},
</math></center>

Similarly, we now consider the potential in the region
<math>\Omega = \left\{ x>r,\;-h\leq z\leq 0\right\} </math>
which satisfies exactly the equations as before except the boundary
condition is
<center><math>
\lim\limits_{x\rightarrow \infty }\phi \left( x,z\right) = \phi_0
T e^{-k_{0}x}
</math></center>
We solve again by separation of variables and obtain
<center><math>
\left. \partial_n \phi \right| _{x=r}=
\left. \partial_x \phi \right| _{x=r}=\mathbf{Q}\left[f\left( z\right)\right] ,
</math></center>
where the outward normal is with respect to the inner domain as before. Note that, if
the depth is different on each side then the matrix <math>\mathbf{Q}</math> will be different
(since we need to use a different depth solving the [[Dispersion Relation for a Free Surface]] etc.)

== Boundary Element Method ==

We have reduced the problem to Laplace's equation in a finite domain subject
to certain boundary conditions. These boundary
conditions give the outward normal derivative of the potential as a function
of the potential but this is not always a point-wise condition; on some
boundaries it is given by an integral equation. We must solve both Laplace's
equation and the integral equations numerically. We will solve Laplace's
equation by the [[:Category:Boundary Element Method|Boundary Element Method]] and the integral equations by
numerical integration. However, the same discretisation of the boundary will
be used for both numerical solutions.
We solve Laplace's equation by a modified constant panel method
which reduces it to the following matrix equation
<center><math>
\frac{1}{2}\vec{\phi}=\partial_n\mathbf{G}\vec{\phi}-\mathbf{G}\partial_n\vec{\phi}.
</math></center>
where <math>\vec{\phi}\mathcal{\ }</math>and <math>\partial_n\vec{\phi}</math>
are vectors which approximate the potential and its normal derivative
around the boundary <math>\partial \Omega </math>, and <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>
are matrices corresponding to the Green function and the outward normal
derivative of the Green function respectively. The method used to calculate
the elements of the matrices <math>\mathbf{G}</math> and <math>\partial_n\mathbf{G}</math> can be
found in [[:Category:Boundary Element Method|Boundary Element Method]].

The outward normal derivative of the potential, <math>\partial_n\vec{\phi},</math> and the
potential, <math>\vec{\phi},</math> are related by the conditions on the boundary <math>
\partial \Omega </math>. This can be expressed as
<center><math>
\partial_n\vec{\phi}=\mathbf{A}\,\vec{\phi}-\vec{f},
</math></center>
where <math>\mathbf{A}</math> is the block diagonal matrix given by
<center><math>
\mathbf{A}\mathbb{=}\left[
\begin{matrix}
\mathbf{Q} & & & \\
& \mathbf{Q} & & \\
& & \alpha \,\mathbf{I} & \\
& & & 0
\end{matrix}
\right] ,
</math></center>
where we have separated the boundaries to the semi-infinite domains, the free surface and the
sea floor and body boundary.
<math>\mathbf{Q}</math> is a matrix
approximation of the integral operator of the same name and <math>\vec{f}</math> is
the vector
<center><math>
\vec{f}=\left[
\begin{matrix}
-2k_{0}\phi_0 \,e^{k_{0}l } \\
0 \\
\vdots \\
0
\end{matrix}
\right] .
</math></center>

We obtain the following matrix equation for the
potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
which can be solved straightforwardly. The reflection and transmission
coefficients are calculated from the solution.

===Numerical Calculation of <math>\mathbf{Q}</math> ===

We begin by truncating to a finite number (<math>N</math>) of
evanescent modes,
<center><math>
\mathbf{Q}\left[ f \right] =-\sum_{m=0}^{N}k_{m}\int_{-h}^{0}
f\left( s\right) \phi_{m}\left( s\right) \mathrm{d}s
\frac{\phi _{m} \left( z\right)}{A_m} .
</math></center>
We calculate the integral
with the same panels as we used to approximate the integrals of the Green
function and its normal derivative . Similarly, we
assume that <math>f(s) \,</math> is constant over each panel and integrate <math>\phi
_{m}\left( s\right) </math> exactly. This gives us the
following matrix factorisation of <math>\mathbf{Q}</math>
<center><math>
\mathbf{Q}[f]=\mathbf{S}\,\mathbf{R}\,[f].
</math></center>
The components of the matrices <math>\mathbf{S}</math> and <math>\mathbf{R}</math> are
<center><math>
s_{jm} = -k_m\phi _{m}\left( z_{j}\right) ,
</math></center>
<center><math>
r_{mj} = \frac{1}{A_m} \int_{z_{j}-\Delta x/2}^{z_{j}+\Delta x/2}\phi
_{m}\left( s\right) \mathrm{d}s
</math></center>

=== Reflection and Transmission Coefficients ===

If we multiply the potential at the left (after subtracting the incident wave) and the right by
<math>\mathbf{R}</math> we can calculate the coefficients in the eigenfunction expansion, and
hence determine the reflection and transmission coefficient.
where <math>z_{j}</math> is the value of the <math>z</math> coordinate in the centre of the panel
and <math>\Delta x</math> is the panel length.

=== Singularity Expansion Method ===
The matrix equation for the potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
can be rewritten as
<center><math>
\mathbf{M}(k) \mathbf{x}=\mathbf{F}(k)
</math></center>
where <math>\mathbf{x}</math> represents the unknown potential values in the new notation. Resonant frequencies (or scattering frequencies) correspond to zero eigenvalues of the matrix <math>\mathbf{M}(k)</math> for a set of discrete points <math>k=k_p</math> in the upper half of the complex plane (due to the time dependence). Since
<center><math>
\mathbf{x}=\left( \mathbf{M}(k)\right)^{-1}\mathbf{F}(k)
</math></center>
then <math>\mathbf{x}</math> will be singular for these scattering frequencies. Each scattering frequency, corresponding to a zero eigenvalue of <math>\mathbf{M}(k)</math> will possess a corresponding eigenvector <math>\mathbf{u}_{k_p}</math> satisfying
<center><math>
\mathbf{M}(k_p)\mathbf{u}_{k_p}=0
</math></center>
and also a related conjugate transpose eigenvector <math>\mathbf{u}^{*}_{k_p}</math> satisfying
<center><math>
\mathbf{M}(k_p)^{A}\mathbf{u}^*_{k_p}=0.
</math></center>
In the vicinity of a scattering frequency the vector of unknowns will satisfy
<center><math>
\mathbf{x}(k)\approx \frac{\langle \mathbf{F}(k),\mathbf{u}^* \rangle}{\langle\mathbf{u},\mathbf{M}'(k)\mathbf{u}^*\rangle}\frac{1}{k-k_p}
</math></center>
and we define a corresponding shape function as
<center><math>
\zeta_{k_p}(x,k_p) =\frac{\langle \mathbf{F}(k_p),\mathbf{u}^* \rangle}{\langle\mathbf{u},\mathbf{M}'(k)\mathbf{u}^*\rangle}\mathbf{x}
</math></center>.

== Matlab Code ==

{{fixed body 2d bem code}}

=== Additional code ===

This program requires
* {{free surface dispersion equation code}}
* {{boundary element code}}

[[Category:Boundary Element Method]]
[[Category:Eigenfunction Matching Method]]

Boundary Element Method for a Fixed Body in Finite Depth

2009-11-24T04:33:01Z

Cfitzgerald: /* Boundary Element Method */

{{complete pages}}

== Introduction ==

We show how to solve for the wave scattering by a rigid body in constant [[Finite Depth]] using
the [[:Category:Boundary Element Method|Boundary Element Method]]. The method can be modified
to account for a body which can move and this is described in
[[Boundary Element Method for a Floating Body in Finite Depth]]

== Equations ==

{{boundary value problem for a fixed body}}

{{incident plane wave}}

{{sommerfeld radiation condition two dimensions}}

== Solution Method ==

We divide the domain into three regions, <math>x<-l</math>,
<math>x>r</math>, and <math>-l<x<r</math> so that the body surface is entirely in the
finite region.

=== Solution in the finite region ===

We use the [[:Category:Boundary Element Method|Boundary Element Method]]
in the finite region.

===Solution in the Semi-infinite Domains===

We now solve Laplace's equation in the semi-infinite domains. First consider
the domain on the left so that <math>\Omega = \left\{ x<-l,\;-h\leq z\leq 0\right\} </math>
The equations are
{{standard linear wave scattering equations without body condition}}
We have the following explicit boundary conditions
<center><math>
\phi =f\left( z\right) ,\;\;x=-\,l
</math></center>
and
<center><math>
\lim\limits_{x\rightarrow -\infty }\phi \left( x,z\right) = \phi_0
e^{-k_{0}x}
+R\phi_0
e^{k_{0}x},
</math></center>
where <math>f\left( z\right) </math> is
an arbitrary continuous function<math>.</math> Our aim is to find the outward normal
derivative of the potential on <math>x=-l</math> as a function of <math>\tilde{\phi}\left(
z\right) </math>. <math>\phi_0</math> and <math>k_0</math> will be defined shortly
when we separate variables, but are equivalent to the [[Sommerfeld Radiation Condition]].

Since the water depth is constant in these regions we can solve Laplace's
equation by separation of variables.

{{separation of variables for a free surface}}

=== Expansion of the potential ===

This gives us the following expression for the potential in the region <math>
\Omega </math>
<center><math>
\phi \left( x,z\right) = \phi_0(z)e^{-k_0 x}
+\sum_{m=0}^{\infty } a_m \phi _{m}\left( z\right)
e^{k_{m}\left( x+l\right) }
</math></center>
where <math>a_m</math> can be found from <math>\left. \phi \right| _{x=-l} = f(z)</math>
from the orthogonality of the vertical eigenfunctions. Therefore
<center><math>
a_m +\delta_{m0}e^{k_0 l} = \frac{1}{A_m} \int_{-h}^{0} f(z) \phi_m(z) \mathrm{d}z
</math></center>

The normal derivative of potential (with the normal outward from the interior region) we obtain
<center><math>
\partial_n\left. \phi \right| _{x=-l}
= -\partial_x \left. \phi \right| _{x=-l}
= k_0 \phi_0(z)e^{k_0 l}
- \sum_{m=0}^{\infty } k_m a_m \phi _{m}\left( z\right)
</math></center>

If we define <math>\mathbf{Q}</math> as
<center><math>
\mathbf{Q} \left[f\left( z\right) \right]=
-\sum_{m=0}^{\infty } \frac{k_{m}}{A_m} \left\{ \int_{-h}^{0} f\left( s\right) \phi
_{m}\left( s\right) \mathrm{d}s \right\} \phi _{m}\left( z\right)
</math></center>
we can write
<center><math>
\partial_n \left. \phi \right| _{x=-l} =\mathbf{Q} \left[ f(z) \right] + 2k_0 \phi_0e^{k_0l},
</math></center>

Similarly, we now consider the potential in the region
<math>\Omega = \left\{ x>r,\;-h\leq z\leq 0\right\} </math>
which satisfies exactly the equations as before except the boundary
condition is
<center><math>
\lim\limits_{x\rightarrow \infty }\phi \left( x,z\right) = \phi_0
T e^{-k_{0}x}
</math></center>
We solve again by separation of variables and obtain
<center><math>
\left. \partial_n \phi \right| _{x=r}=
\left. \partial_x \phi \right| _{x=r}=\mathbf{Q}\left[f\left( z\right)\right] ,
</math></center>
where the outward normal is with respect to the inner domain as before. Note that, if
the depth is different on each side then the matrix <math>\mathbf{Q}</math> will be different
(since we need to use a different depth solving the [[Dispersion Relation for a Free Surface]] etc.)

== Boundary Element Method ==

We have reduced the problem to Laplace's equation in a finite domain subject
to certain boundary conditions. These boundary
conditions give the outward normal derivative of the potential as a function
of the potential but this is not always a point-wise condition; on some
boundaries it is given by an integral equation. We must solve both Laplace's
equation and the integral equations numerically. We will solve Laplace's
equation by the [[:Category:Boundary Element Method|Boundary Element Method]] and the integral equations by
numerical integration. However, the same discretisation of the boundary will
be used for both numerical solutions.
We solve Laplace's equation by a modified constant panel method
which reduces it to the following matrix equation
<center><math>
\frac{1}{2}\vec{\phi}=\partial_n\mathbf{G}\vec{\phi}-\mathbf{G}\partial_n\vec{\phi}.
</math></center>
where <math>\vec{\phi}\mathcal{\ }</math>and <math>\partial_n\vec{\phi}</math>
are vectors which approximate the potential and its normal derivative
around the boundary <math>\partial \Omega </math>, and <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>
are matrices corresponding to the Green function and the outward normal
derivative of the Green function respectively. The method used to calculate
the elements of the matrices <math>\mathbf{G}</math> and <math>\partial_n\mathbf{G}</math> can be
found in [[:Category:Boundary Element Method|Boundary Element Method]].

The outward normal derivative of the potential, <math>\partial_n\vec{\phi},</math> and the
potential, <math>\vec{\phi},</math> are related by the conditions on the boundary <math>
\partial \Omega </math>. This can be expressed as
<center><math>
\partial_n\vec{\phi}=\mathbf{A}\,\vec{\phi}-\vec{f},
</math></center>
where <math>\mathbf{A}</math> is the block diagonal matrix given by
<center><math>
\mathbf{A}\mathbb{=}\left[
\begin{matrix}
\mathbf{Q} & & & \\
& \mathbf{Q} & & \\
& & \alpha \,\mathbf{I} & \\
& & & 0
\end{matrix}
\right] ,
</math></center>
where we have separated the boundaries to the semi-infinite domains, the free surface and the
sea floor and body boundary.
<math>\mathbf{Q}</math> is a matrix
approximation of the integral operator of the same name and <math>\vec{f}</math> is
the vector
<center><math>
\vec{f}=\left[
\begin{matrix}
-2k_{0}\phi_0 \,e^{k_{0}l } \\
0 \\
\vdots \\
0
\end{matrix}
\right] .
</math></center>

We obtain the following matrix equation for the
potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
which can be solved straightforwardly. The reflection and transmission
coefficients are calculated from the solution.

===Numerical Calculation of <math>\mathbf{Q}</math> ===

We begin by truncating to a finite number (<math>N</math>) of
evanescent modes,
<center><math>
\mathbf{Q}\left[ f \right] =-\sum_{m=0}^{N}k_{m}\int_{-h}^{0}
f\left( s\right) \phi_{m}\left( s\right) \mathrm{d}s
\frac{\phi _{m} \left( z\right)}{A_m} .
</math></center>
We calculate the integral
with the same panels as we used to approximate the integrals of the Green
function and its normal derivative . Similarly, we
assume that <math>f(s) \,</math> is constant over each panel and integrate <math>\phi
_{m}\left( s\right) </math> exactly. This gives us the
following matrix factorisation of <math>\mathbf{Q}</math>
<center><math>
\mathbf{Q}[f]=\mathbf{S}\,\mathbf{R}\,[f].
</math></center>
The components of the matrices <math>\mathbf{S}</math> and <math>\mathbf{R}</math> are
<center><math>
s_{jm} = -k_m\phi _{m}\left( z_{j}\right) ,
</math></center>
<center><math>
r_{mj} = \frac{1}{A_m} \int_{z_{j}-\Delta x/2}^{z_{j}+\Delta x/2}\phi
_{m}\left( s\right) \mathrm{d}s
</math></center>

=== Reflection and Transmission Coefficients ===

If we multiply the potential at the left (after subtracting the incident wave) and the right by
<math>\mathbf{R}</math> we can calculate the coefficients in the eigenfunction expansion, and
hence determine the reflection and transmission coefficient.
where <math>z_{j}</math> is the value of the <math>z</math> coordinate in the centre of the panel
and <math>\Delta x</math> is the panel length.

=== Singularity Expansion Method ===
The matrix equation for the potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
can be rewritten as
<center><math>
\mathbf{M}(k) \mathbf{x}=\mathbf{F}(k)
</math></center>
where <math>\mathbf{x}</math> represents the unknown potential values in the new notation. Resonant frequencies (or scattering frequencies) correspond to zero eigenvalues of the matrix <math>\mathbf{M}(k)</math> for a set of discrete points <math>k=k_p</math> in the upper half of the complex plane (due to the time dependence). Since
<center><math>
\mathbf{x}=\left( \mathbf{M}(k)\right)^{-1}\mathbf{F}(k)
</math></center>
then <math>\mathbf{x}</math> will be singular for these scattering frequencies. Each scattering frequency, corresponding to a zero eigenvalue of <math>\mathbf{M}(k)</math> will possess a corresponding eigenvector <math>\mathbf{u}_{k_p}</math> satisfying
<center><math>
\mathbf{M}(k_p)\mathbf{u}_{k_p}=0
</math></center>
and also a related conjugate transpose eigenvector <math>\mathbf{u}^{*}_{k_p}</math> satisfying
<center><math>
\mathbf{M}(k_p)^{A}\mathbf{u}^*_{k_p}=0.
</math></center>
In the vicinity of a scattering frequency the vector of unknowns will satisfy
<center><math>
\mathbf{x}(k)\approx \frac{\langle \mathbf{F}(k),\mathbf{u}^* \rangle}{\langle\mathbf{u},\mathbf{M}'(k)\mathbf{u}^*\rangle}\frac{1}{k-k_p}
</math></center>
and we define a corresponding shape function as
<center><math>
\zeta_{k_p}(x,k_p) =\frac{\langle \mathbf{F}(k_p),\mathbf{u}^* \rangle}{\langle\mathbf{u},\mathbf{M}'(k)\mathbf{u}^*\rangle}\mathbf{x}
</math></center>

== Matlab Code ==

{{fixed body 2d bem code}}

=== Additional code ===

This program requires
* {{free surface dispersion equation code}}
* {{boundary element code}}

[[Category:Boundary Element Method]]
[[Category:Eigenfunction Matching Method]]

Boundary Element Method for a Fixed Body in Finite Depth

2009-11-24T04:29:47Z

Cfitzgerald: /* Boundary Element Method */

{{complete pages}}

== Introduction ==

We show how to solve for the wave scattering by a rigid body in constant [[Finite Depth]] using
the [[:Category:Boundary Element Method|Boundary Element Method]]. The method can be modified
to account for a body which can move and this is described in
[[Boundary Element Method for a Floating Body in Finite Depth]]

== Equations ==

{{boundary value problem for a fixed body}}

{{incident plane wave}}

{{sommerfeld radiation condition two dimensions}}

== Solution Method ==

We divide the domain into three regions, <math>x<-l</math>,
<math>x>r</math>, and <math>-l<x<r</math> so that the body surface is entirely in the
finite region.

=== Solution in the finite region ===

We use the [[:Category:Boundary Element Method|Boundary Element Method]]
in the finite region.

===Solution in the Semi-infinite Domains===

We now solve Laplace's equation in the semi-infinite domains. First consider
the domain on the left so that <math>\Omega = \left\{ x<-l,\;-h\leq z\leq 0\right\} </math>
The equations are
{{standard linear wave scattering equations without body condition}}
We have the following explicit boundary conditions
<center><math>
\phi =f\left( z\right) ,\;\;x=-\,l
</math></center>
and
<center><math>
\lim\limits_{x\rightarrow -\infty }\phi \left( x,z\right) = \phi_0
e^{-k_{0}x}
+R\phi_0
e^{k_{0}x},
</math></center>
where <math>f\left( z\right) </math> is
an arbitrary continuous function<math>.</math> Our aim is to find the outward normal
derivative of the potential on <math>x=-l</math> as a function of <math>\tilde{\phi}\left(
z\right) </math>. <math>\phi_0</math> and <math>k_0</math> will be defined shortly
when we separate variables, but are equivalent to the [[Sommerfeld Radiation Condition]].

Since the water depth is constant in these regions we can solve Laplace's
equation by separation of variables.

{{separation of variables for a free surface}}

=== Expansion of the potential ===

This gives us the following expression for the potential in the region <math>
\Omega </math>
<center><math>
\phi \left( x,z\right) = \phi_0(z)e^{-k_0 x}
+\sum_{m=0}^{\infty } a_m \phi _{m}\left( z\right)
e^{k_{m}\left( x+l\right) }
</math></center>
where <math>a_m</math> can be found from <math>\left. \phi \right| _{x=-l} = f(z)</math>
from the orthogonality of the vertical eigenfunctions. Therefore
<center><math>
a_m +\delta_{m0}e^{k_0 l} = \frac{1}{A_m} \int_{-h}^{0} f(z) \phi_m(z) \mathrm{d}z
</math></center>

The normal derivative of potential (with the normal outward from the interior region) we obtain
<center><math>
\partial_n\left. \phi \right| _{x=-l}
= -\partial_x \left. \phi \right| _{x=-l}
= k_0 \phi_0(z)e^{k_0 l}
- \sum_{m=0}^{\infty } k_m a_m \phi _{m}\left( z\right)
</math></center>

If we define <math>\mathbf{Q}</math> as
<center><math>
\mathbf{Q} \left[f\left( z\right) \right]=
-\sum_{m=0}^{\infty } \frac{k_{m}}{A_m} \left\{ \int_{-h}^{0} f\left( s\right) \phi
_{m}\left( s\right) \mathrm{d}s \right\} \phi _{m}\left( z\right)
</math></center>
we can write
<center><math>
\partial_n \left. \phi \right| _{x=-l} =\mathbf{Q} \left[ f(z) \right] + 2k_0 \phi_0e^{k_0l},
</math></center>

Similarly, we now consider the potential in the region
<math>\Omega = \left\{ x>r,\;-h\leq z\leq 0\right\} </math>
which satisfies exactly the equations as before except the boundary
condition is
<center><math>
\lim\limits_{x\rightarrow \infty }\phi \left( x,z\right) = \phi_0
T e^{-k_{0}x}
</math></center>
We solve again by separation of variables and obtain
<center><math>
\left. \partial_n \phi \right| _{x=r}=
\left. \partial_x \phi \right| _{x=r}=\mathbf{Q}\left[f\left( z\right)\right] ,
</math></center>
where the outward normal is with respect to the inner domain as before. Note that, if
the depth is different on each side then the matrix <math>\mathbf{Q}</math> will be different
(since we need to use a different depth solving the [[Dispersion Relation for a Free Surface]] etc.)

== Boundary Element Method ==

We have reduced the problem to Laplace's equation in a finite domain subject
to certain boundary conditions. These boundary
conditions give the outward normal derivative of the potential as a function
of the potential but this is not always a point-wise condition; on some
boundaries it is given by an integral equation. We must solve both Laplace's
equation and the integral equations numerically. We will solve Laplace's
equation by the [[:Category:Boundary Element Method|Boundary Element Method]] and the integral equations by
numerical integration. However, the same discretisation of the boundary will
be used for both numerical solutions.
We solve Laplace's equation by a modified constant panel method
which reduces it to the following matrix equation
<center><math>
\frac{1}{2}\vec{\phi}=\partial_n\mathbf{G}\vec{\phi}-\mathbf{G}\partial_n\vec{\phi}.
</math></center>
where <math>\vec{\phi}\mathcal{\ }</math>and <math>\partial_n\vec{\phi}</math>
are vectors which approximate the potential and its normal derivative
around the boundary <math>\partial \Omega </math>, and <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>
are matrices corresponding to the Green function and the outward normal
derivative of the Green function respectively. The method used to calculate
the elements of the matrices <math>\mathbf{G}</math> and <math>\partial_n\mathbf{G}</math> can be
found in [[:Category:Boundary Element Method|Boundary Element Method]].

The outward normal derivative of the potential, <math>\partial_n\vec{\phi},</math> and the
potential, <math>\vec{\phi},</math> are related by the conditions on the boundary <math>
\partial \Omega </math>. This can be expressed as
<center><math>
\partial_n\vec{\phi}=\mathbf{A}\,\vec{\phi}-\vec{f},
</math></center>
where <math>\mathbf{A}</math> is the block diagonal matrix given by
<center><math>
\mathbf{A}\mathbb{=}\left[
\begin{matrix}
\mathbf{Q} & & & \\
& \mathbf{Q} & & \\
& & \alpha \,\mathbf{I} & \\
& & & 0
\end{matrix}
\right] ,
</math></center>
where we have separated the boundaries to the semi-infinite domains, the free surface and the
sea floor and body boundary.
<math>\mathbf{Q}</math> is a matrix
approximation of the integral operator of the same name and <math>\vec{f}</math> is
the vector
<center><math>
\vec{f}=\left[
\begin{matrix}
-2k_{0}\phi_0 \,e^{k_{0}l } \\
0 \\
\vdots \\
0
\end{matrix}
\right] .
</math></center>

We obtain the following matrix equation for the
potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
which can be solved straightforwardly. The reflection and transmission
coefficients are calculated from the solution.

===Numerical Calculation of <math>\mathbf{Q}</math> ===

We begin by truncating to a finite number (<math>N</math>) of
evanescent modes,
<center><math>
\mathbf{Q}\left[ f \right] =-\sum_{m=0}^{N}k_{m}\int_{-h}^{0}
f\left( s\right) \phi_{m}\left( s\right) \mathrm{d}s
\frac{\phi _{m} \left( z\right)}{A_m} .
</math></center>
We calculate the integral
with the same panels as we used to approximate the integrals of the Green
function and its normal derivative . Similarly, we
assume that <math>f(s) \,</math> is constant over each panel and integrate <math>\phi
_{m}\left( s\right) </math> exactly. This gives us the
following matrix factorisation of <math>\mathbf{Q}</math>
<center><math>
\mathbf{Q}[f]=\mathbf{S}\,\mathbf{R}\,[f].
</math></center>
The components of the matrices <math>\mathbf{S}</math> and <math>\mathbf{R}</math> are
<center><math>
s_{jm} = -k_m\phi _{m}\left( z_{j}\right) ,
</math></center>
<center><math>
r_{mj} = \frac{1}{A_m} \int_{z_{j}-\Delta x/2}^{z_{j}+\Delta x/2}\phi
_{m}\left( s\right) \mathrm{d}s
</math></center>

=== Reflection and Transmission Coefficients ===

If we multiply the potential at the left (after subtracting the incident wave) and the right by
<math>\mathbf{R}</math> we can calculate the coefficients in the eigenfunction expansion, and
hence determine the reflection and transmission coefficient.
where <math>z_{j}</math> is the value of the <math>z</math> coordinate in the centre of the panel
and <math>\Delta x</math> is the panel length.

=== Singularity Expansion Method ===
The matrix equation for the potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
can be rewritten as
<center><math>
\mathbf{M}(k) \mathbf{x}=\mathbf{F}(k)
</math></center>
where <math>\mathbf{x}</math> represents the unknown potential values in the new notation. Resonant frequencies (or scattering frequencies) correspond to zero eigenvalues of the matrix <math>\mathbf{M}(k)</math> for a set of discrete points <math>k=k_p</math> in the upper half of the complex plane (due to the time dependence). Since
<center><math>
\mathbf{x}=\left( \mathbf{M}(k)\right)^{-1}\mathbf{F}(k)
</math></center>
then <math>\mathbf{x}</math> will be singular for these scattering frequencies. Each scattering frequency, corresponding to a zero eigenvalue of <math>\mathbf{M}(k)</math> will possess a corresponding eigenvector <math>\mathbf{u}_{k_p}</math> satisfying
<center><math>
\mathbf{M}(k_p)\mathbf{u}_{k_p}=0
</math></center>
and also a related conjugate transpose eigenvector <math>\mathbf{u}^{*}_{k_p}</math> satisfying
<center><math>
\mathbf{M}(k_p)^{A}\mathbf{u}^*_{k_p}=0.
</math></center>
In the vicinity of a scattering frequency the vector of unknowns will satisfy
<center><math>
\mathbf{x}(k)\approx \frac{\langle \mathbf{F},\mathbf{u}^* \rangle}{\langle\mathbf{u},\mathbf{M}'(k)\mathbf{u}^*\rangle}
</math></center>

== Matlab Code ==

{{fixed body 2d bem code}}

=== Additional code ===

This program requires
* {{free surface dispersion equation code}}
* {{boundary element code}}

[[Category:Boundary Element Method]]
[[Category:Eigenfunction Matching Method]]

Boundary Element Method for a Fixed Body in Finite Depth

2009-11-24T04:28:57Z

Cfitzgerald: /* Boundary Element Method */

{{complete pages}}

== Introduction ==

We show how to solve for the wave scattering by a rigid body in constant [[Finite Depth]] using
the [[:Category:Boundary Element Method|Boundary Element Method]]. The method can be modified
to account for a body which can move and this is described in
[[Boundary Element Method for a Floating Body in Finite Depth]]

== Equations ==

{{boundary value problem for a fixed body}}

{{incident plane wave}}

{{sommerfeld radiation condition two dimensions}}

== Solution Method ==

We divide the domain into three regions, <math>x<-l</math>,
<math>x>r</math>, and <math>-l<x<r</math> so that the body surface is entirely in the
finite region.

=== Solution in the finite region ===

We use the [[:Category:Boundary Element Method|Boundary Element Method]]
in the finite region.

===Solution in the Semi-infinite Domains===

We now solve Laplace's equation in the semi-infinite domains. First consider
the domain on the left so that <math>\Omega = \left\{ x<-l,\;-h\leq z\leq 0\right\} </math>
The equations are
{{standard linear wave scattering equations without body condition}}
We have the following explicit boundary conditions
<center><math>
\phi =f\left( z\right) ,\;\;x=-\,l
</math></center>
and
<center><math>
\lim\limits_{x\rightarrow -\infty }\phi \left( x,z\right) = \phi_0
e^{-k_{0}x}
+R\phi_0
e^{k_{0}x},
</math></center>
where <math>f\left( z\right) </math> is
an arbitrary continuous function<math>.</math> Our aim is to find the outward normal
derivative of the potential on <math>x=-l</math> as a function of <math>\tilde{\phi}\left(
z\right) </math>. <math>\phi_0</math> and <math>k_0</math> will be defined shortly
when we separate variables, but are equivalent to the [[Sommerfeld Radiation Condition]].

Since the water depth is constant in these regions we can solve Laplace's
equation by separation of variables.

{{separation of variables for a free surface}}

=== Expansion of the potential ===

This gives us the following expression for the potential in the region <math>
\Omega </math>
<center><math>
\phi \left( x,z\right) = \phi_0(z)e^{-k_0 x}
+\sum_{m=0}^{\infty } a_m \phi _{m}\left( z\right)
e^{k_{m}\left( x+l\right) }
</math></center>
where <math>a_m</math> can be found from <math>\left. \phi \right| _{x=-l} = f(z)</math>
from the orthogonality of the vertical eigenfunctions. Therefore
<center><math>
a_m +\delta_{m0}e^{k_0 l} = \frac{1}{A_m} \int_{-h}^{0} f(z) \phi_m(z) \mathrm{d}z
</math></center>

The normal derivative of potential (with the normal outward from the interior region) we obtain
<center><math>
\partial_n\left. \phi \right| _{x=-l}
= -\partial_x \left. \phi \right| _{x=-l}
= k_0 \phi_0(z)e^{k_0 l}
- \sum_{m=0}^{\infty } k_m a_m \phi _{m}\left( z\right)
</math></center>

If we define <math>\mathbf{Q}</math> as
<center><math>
\mathbf{Q} \left[f\left( z\right) \right]=
-\sum_{m=0}^{\infty } \frac{k_{m}}{A_m} \left\{ \int_{-h}^{0} f\left( s\right) \phi
_{m}\left( s\right) \mathrm{d}s \right\} \phi _{m}\left( z\right)
</math></center>
we can write
<center><math>
\partial_n \left. \phi \right| _{x=-l} =\mathbf{Q} \left[ f(z) \right] + 2k_0 \phi_0e^{k_0l},
</math></center>

Similarly, we now consider the potential in the region
<math>\Omega = \left\{ x>r,\;-h\leq z\leq 0\right\} </math>
which satisfies exactly the equations as before except the boundary
condition is
<center><math>
\lim\limits_{x\rightarrow \infty }\phi \left( x,z\right) = \phi_0
T e^{-k_{0}x}
</math></center>
We solve again by separation of variables and obtain
<center><math>
\left. \partial_n \phi \right| _{x=r}=
\left. \partial_x \phi \right| _{x=r}=\mathbf{Q}\left[f\left( z\right)\right] ,
</math></center>
where the outward normal is with respect to the inner domain as before. Note that, if
the depth is different on each side then the matrix <math>\mathbf{Q}</math> will be different
(since we need to use a different depth solving the [[Dispersion Relation for a Free Surface]] etc.)

== Boundary Element Method ==

We have reduced the problem to Laplace's equation in a finite domain subject
to certain boundary conditions. These boundary
conditions give the outward normal derivative of the potential as a function
of the potential but this is not always a point-wise condition; on some
boundaries it is given by an integral equation. We must solve both Laplace's
equation and the integral equations numerically. We will solve Laplace's
equation by the [[:Category:Boundary Element Method|Boundary Element Method]] and the integral equations by
numerical integration. However, the same discretisation of the boundary will
be used for both numerical solutions.
We solve Laplace's equation by a modified constant panel method
which reduces it to the following matrix equation
<center><math>
\frac{1}{2}\vec{\phi}=\partial_n\mathbf{G}\vec{\phi}-\mathbf{G}\partial_n\vec{\phi}.
</math></center>
where <math>\vec{\phi}\mathcal{\ }</math>and <math>\partial_n\vec{\phi}</math>
are vectors which approximate the potential and its normal derivative
around the boundary <math>\partial \Omega </math>, and <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>
are matrices corresponding to the Green function and the outward normal
derivative of the Green function respectively. The method used to calculate
the elements of the matrices <math>\mathbf{G}</math> and <math>\partial_n\mathbf{G}</math> can be
found in [[:Category:Boundary Element Method|Boundary Element Method]].

The outward normal derivative of the potential, <math>\partial_n\vec{\phi},</math> and the
potential, <math>\vec{\phi},</math> are related by the conditions on the boundary <math>
\partial \Omega </math>. This can be expressed as
<center><math>
\partial_n\vec{\phi}=\mathbf{A}\,\vec{\phi}-\vec{f},
</math></center>
where <math>\mathbf{A}</math> is the block diagonal matrix given by
<center><math>
\mathbf{A}\mathbb{=}\left[
\begin{matrix}
\mathbf{Q} & & & \\
& \mathbf{Q} & & \\
& & \alpha \,\mathbf{I} & \\
& & & 0
\end{matrix}
\right] ,
</math></center>
where we have separated the boundaries to the semi-infinite domains, the free surface and the
sea floor and body boundary.
<math>\mathbf{Q}</math> is a matrix
approximation of the integral operator of the same name and <math>\vec{f}</math> is
the vector
<center><math>
\vec{f}=\left[
\begin{matrix}
-2k_{0}\phi_0 \,e^{k_{0}l } \\
0 \\
\vdots \\
0
\end{matrix}
\right] .
</math></center>

We obtain the following matrix equation for the
potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
which can be solved straightforwardly. The reflection and transmission
coefficients are calculated from the solution.

===Numerical Calculation of <math>\mathbf{Q}</math> ===

We begin by truncating to a finite number (<math>N</math>) of
evanescent modes,
<center><math>
\mathbf{Q}\left[ f \right] =-\sum_{m=0}^{N}k_{m}\int_{-h}^{0}
f\left( s\right) \phi_{m}\left( s\right) \mathrm{d}s
\frac{\phi _{m} \left( z\right)}{A_m} .
</math></center>
We calculate the integral
with the same panels as we used to approximate the integrals of the Green
function and its normal derivative . Similarly, we
assume that <math>f(s) \,</math> is constant over each panel and integrate <math>\phi
_{m}\left( s\right) </math> exactly. This gives us the
following matrix factorisation of <math>\mathbf{Q}</math>
<center><math>
\mathbf{Q}[f]=\mathbf{S}\,\mathbf{R}\,[f].
</math></center>
The components of the matrices <math>\mathbf{S}</math> and <math>\mathbf{R}</math> are
<center><math>
s_{jm} = -k_m\phi _{m}\left( z_{j}\right) ,
</math></center>
<center><math>
r_{mj} = \frac{1}{A_m} \int_{z_{j}-\Delta x/2}^{z_{j}+\Delta x/2}\phi
_{m}\left( s\right) \mathrm{d}s
</math></center>

=== Reflection and Transmission Coefficients ===

If we multiply the potential at the left (after subtracting the incident wave) and the right by
<math>\mathbf{R}</math> we can calculate the coefficients in the eigenfunction expansion, and
hence determine the reflection and transmission coefficient.
where <math>z_{j}</math> is the value of the <math>z</math> coordinate in the centre of the panel
and <math>\Delta x</math> is the panel length.

=== Singularity Expansion Method ===
The matrix equation for the potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
can be rewritten as
<center><math>
\mathbf{M}(k) \mathbf{x}=\mathbf{F}(k)
</math></center>
where <math>\mathbf{x}</math> represents the unknown potential values in the new notation. Resonant frequencies (or scattering frequencies) correspond to zero eigenvalues of the matrix <math>\mathbf{M}(k)</math> for a set of discrete points <math>k=k_p</math> in the upper half of the complex plane (due to the time dependence). Since
<center><math>
\mathbf{x}=\left( \mathbf{M}(k)\right)^{-1}\mathbf{F}(k)
</math></center>
then <math>\mathbf{x}</math> will be singular for these scattering frequencies. Each scattering frequency, corresponding to a zero eigenvalue of <math>\mathbf{M}(k)</math> will possess a corresponding eigenvector <math>\mathbf{u}_{k_p}<\math> satisfying
<center><math>
\mathbf{M}(k_p)\mathbf{u}_{k_p}=0
</math></center>
and also a related conjugate transpose eigenvector <math>\mathbf{u}^{*}_{k_p}<\math> satisfying
<center><math>
\mathbf{M}(k_p)^{A}\mathbf{u}^*_{k_p}=0.
</math></center>
In the vicinity of a scattering frequency the vector of unknowns will satisfy
<center><math>
\mathbf{x}(k)\approx \frac{\langle \mathbf{F},\mathbf{u}^* \rangle}{\langle\mathbf{u},\mathbf{M}'(k)\mathbf{u}^*\rangle}
</math></center>

== Matlab Code ==

{{fixed body 2d bem code}}

=== Additional code ===

This program requires
* {{free surface dispersion equation code}}
* {{boundary element code}}

[[Category:Boundary Element Method]]
[[Category:Eigenfunction Matching Method]]

Boundary Element Method for a Fixed Body in Finite Depth

2009-11-24T04:22:51Z

Cfitzgerald: /* Boundary Element Method */

{{complete pages}}

== Introduction ==

We show how to solve for the wave scattering by a rigid body in constant [[Finite Depth]] using
the [[:Category:Boundary Element Method|Boundary Element Method]]. The method can be modified
to account for a body which can move and this is described in
[[Boundary Element Method for a Floating Body in Finite Depth]]

== Equations ==

{{boundary value problem for a fixed body}}

{{incident plane wave}}

{{sommerfeld radiation condition two dimensions}}

== Solution Method ==

We divide the domain into three regions, <math>x<-l</math>,
<math>x>r</math>, and <math>-l<x<r</math> so that the body surface is entirely in the
finite region.

=== Solution in the finite region ===

We use the [[:Category:Boundary Element Method|Boundary Element Method]]
in the finite region.

===Solution in the Semi-infinite Domains===

We now solve Laplace's equation in the semi-infinite domains. First consider
the domain on the left so that <math>\Omega = \left\{ x<-l,\;-h\leq z\leq 0\right\} </math>
The equations are
{{standard linear wave scattering equations without body condition}}
We have the following explicit boundary conditions
<center><math>
\phi =f\left( z\right) ,\;\;x=-\,l
</math></center>
and
<center><math>
\lim\limits_{x\rightarrow -\infty }\phi \left( x,z\right) = \phi_0
e^{-k_{0}x}
+R\phi_0
e^{k_{0}x},
</math></center>
where <math>f\left( z\right) </math> is
an arbitrary continuous function<math>.</math> Our aim is to find the outward normal
derivative of the potential on <math>x=-l</math> as a function of <math>\tilde{\phi}\left(
z\right) </math>. <math>\phi_0</math> and <math>k_0</math> will be defined shortly
when we separate variables, but are equivalent to the [[Sommerfeld Radiation Condition]].

Since the water depth is constant in these regions we can solve Laplace's
equation by separation of variables.

{{separation of variables for a free surface}}

=== Expansion of the potential ===

This gives us the following expression for the potential in the region <math>
\Omega </math>
<center><math>
\phi \left( x,z\right) = \phi_0(z)e^{-k_0 x}
+\sum_{m=0}^{\infty } a_m \phi _{m}\left( z\right)
e^{k_{m}\left( x+l\right) }
</math></center>
where <math>a_m</math> can be found from <math>\left. \phi \right| _{x=-l} = f(z)</math>
from the orthogonality of the vertical eigenfunctions. Therefore
<center><math>
a_m +\delta_{m0}e^{k_0 l} = \frac{1}{A_m} \int_{-h}^{0} f(z) \phi_m(z) \mathrm{d}z
</math></center>

The normal derivative of potential (with the normal outward from the interior region) we obtain
<center><math>
\partial_n\left. \phi \right| _{x=-l}
= -\partial_x \left. \phi \right| _{x=-l}
= k_0 \phi_0(z)e^{k_0 l}
- \sum_{m=0}^{\infty } k_m a_m \phi _{m}\left( z\right)
</math></center>

If we define <math>\mathbf{Q}</math> as
<center><math>
\mathbf{Q} \left[f\left( z\right) \right]=
-\sum_{m=0}^{\infty } \frac{k_{m}}{A_m} \left\{ \int_{-h}^{0} f\left( s\right) \phi
_{m}\left( s\right) \mathrm{d}s \right\} \phi _{m}\left( z\right)
</math></center>
we can write
<center><math>
\partial_n \left. \phi \right| _{x=-l} =\mathbf{Q} \left[ f(z) \right] + 2k_0 \phi_0e^{k_0l},
</math></center>

Similarly, we now consider the potential in the region
<math>\Omega = \left\{ x>r,\;-h\leq z\leq 0\right\} </math>
which satisfies exactly the equations as before except the boundary
condition is
<center><math>
\lim\limits_{x\rightarrow \infty }\phi \left( x,z\right) = \phi_0
T e^{-k_{0}x}
</math></center>
We solve again by separation of variables and obtain
<center><math>
\left. \partial_n \phi \right| _{x=r}=
\left. \partial_x \phi \right| _{x=r}=\mathbf{Q}\left[f\left( z\right)\right] ,
</math></center>
where the outward normal is with respect to the inner domain as before. Note that, if
the depth is different on each side then the matrix <math>\mathbf{Q}</math> will be different
(since we need to use a different depth solving the [[Dispersion Relation for a Free Surface]] etc.)

== Boundary Element Method ==

We have reduced the problem to Laplace's equation in a finite domain subject
to certain boundary conditions. These boundary
conditions give the outward normal derivative of the potential as a function
of the potential but this is not always a point-wise condition; on some
boundaries it is given by an integral equation. We must solve both Laplace's
equation and the integral equations numerically. We will solve Laplace's
equation by the [[:Category:Boundary Element Method|Boundary Element Method]] and the integral equations by
numerical integration. However, the same discretisation of the boundary will
be used for both numerical solutions.
We solve Laplace's equation by a modified constant panel method
which reduces it to the following matrix equation
<center><math>
\frac{1}{2}\vec{\phi}=\partial_n\mathbf{G}\vec{\phi}-\mathbf{G}\partial_n\vec{\phi}.
</math></center>
where <math>\vec{\phi}\mathcal{\ }</math>and <math>\partial_n\vec{\phi}</math>
are vectors which approximate the potential and its normal derivative
around the boundary <math>\partial \Omega </math>, and <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>
are matrices corresponding to the Green function and the outward normal
derivative of the Green function respectively. The method used to calculate
the elements of the matrices <math>\mathbf{G}</math> and <math>\partial_n\mathbf{G}</math> can be
found in [[:Category:Boundary Element Method|Boundary Element Method]].

The outward normal derivative of the potential, <math>\partial_n\vec{\phi},</math> and the
potential, <math>\vec{\phi},</math> are related by the conditions on the boundary <math>
\partial \Omega </math>. This can be expressed as
<center><math>
\partial_n\vec{\phi}=\mathbf{A}\,\vec{\phi}-\vec{f},
</math></center>
where <math>\mathbf{A}</math> is the block diagonal matrix given by
<center><math>
\mathbf{A}\mathbb{=}\left[
\begin{matrix}
\mathbf{Q} & & & \\
& \mathbf{Q} & & \\
& & \alpha \,\mathbf{I} & \\
& & & 0
\end{matrix}
\right] ,
</math></center>
where we have separated the boundaries to the semi-infinite domains, the free surface and the
sea floor and body boundary.
<math>\mathbf{Q}</math> is a matrix
approximation of the integral operator of the same name and <math>\vec{f}</math> is
the vector
<center><math>
\vec{f}=\left[
\begin{matrix}
-2k_{0}\phi_0 \,e^{k_{0}l } \\
0 \\
\vdots \\
0
\end{matrix}
\right] .
</math></center>

We obtain the following matrix equation for the
potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
which can be solved straightforwardly. The reflection and transmission
coefficients are calculated from the solution.

===Numerical Calculation of <math>\mathbf{Q}</math> ===

We begin by truncating to a finite number (<math>N</math>) of
evanescent modes,
<center><math>
\mathbf{Q}\left[ f \right] =-\sum_{m=0}^{N}k_{m}\int_{-h}^{0}
f\left( s\right) \phi_{m}\left( s\right) \mathrm{d}s
\frac{\phi _{m} \left( z\right)}{A_m} .
</math></center>
We calculate the integral
with the same panels as we used to approximate the integrals of the Green
function and its normal derivative . Similarly, we
assume that <math>f(s) \,</math> is constant over each panel and integrate <math>\phi
_{m}\left( s\right) </math> exactly. This gives us the
following matrix factorisation of <math>\mathbf{Q}</math>
<center><math>
\mathbf{Q}[f]=\mathbf{S}\,\mathbf{R}\,[f].
</math></center>
The components of the matrices <math>\mathbf{S}</math> and <math>\mathbf{R}</math> are
<center><math>
s_{jm} = -k_m\phi _{m}\left( z_{j}\right) ,
</math></center>
<center><math>
r_{mj} = \frac{1}{A_m} \int_{z_{j}-\Delta x/2}^{z_{j}+\Delta x/2}\phi
_{m}\left( s\right) \mathrm{d}s
</math></center>

=== Reflection and Transmission Coefficients ===

If we multiply the potential at the left (after subtracting the incident wave) and the right by
<math>\mathbf{R}</math> we can calculate the coefficients in the eigenfunction expansion, and
hence determine the reflection and transmission coefficient.
where <math>z_{j}</math> is the value of the <math>z</math> coordinate in the centre of the panel
and <math>\Delta x</math> is the panel length.

=== Singularity Expansion Method ===
The matrix equation for the potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
can be rewritten as
<center><math>
\mathbf{M}(k) \vec{x}=\vec{F}(k)
</math></center>
where <math>\vec{x}</math> represents the unknown potential values in the new notation. Resonant frequencies (or scattering frequencies) correspond to zero eigenvalues of the matrix <math>\mathbf{M}(k)</math> for a set of discrete points <math>k=k_p</math> in the upper half of the complex plane (due to the time dependence). Since
<center><math>
\vec{x}=\left( \mathbf{M}(k)\right)^{-1}\vec{F}(k)
</math></center>
then <math>\vec{x}</math> will be singular for these scattering frequencies. Each scattering frequency, corresponding to a zero eigenvalue of <math>\mathbf{M}(k)</math> will possess a corresponding eigenvector <math>\mathbf{u}_{k_p}<\math> satisfying
<center><math>
\mathbf{M}(k_p)\mathbf{u}_{k_p}=0
</math></center>
and also a related conjugate transpose eigenvector <math>\mathbf{u}^{*}_{k_p}<\math> satisfying
<center><math>
\mathbf{M}(k_p)\mathbf{u}_{k_p}=0
</math></center>

== Matlab Code ==

{{fixed body 2d bem code}}

=== Additional code ===

This program requires
* {{free surface dispersion equation code}}
* {{boundary element code}}

[[Category:Boundary Element Method]]
[[Category:Eigenfunction Matching Method]]

Boundary Element Method for a Fixed Body in Finite Depth

2009-11-24T04:22:08Z

Cfitzgerald: /* Boundary Element Method */

{{complete pages}}

== Introduction ==

We show how to solve for the wave scattering by a rigid body in constant [[Finite Depth]] using
the [[:Category:Boundary Element Method|Boundary Element Method]]. The method can be modified
to account for a body which can move and this is described in
[[Boundary Element Method for a Floating Body in Finite Depth]]

== Equations ==

{{boundary value problem for a fixed body}}

{{incident plane wave}}

{{sommerfeld radiation condition two dimensions}}

== Solution Method ==

We divide the domain into three regions, <math>x<-l</math>,
<math>x>r</math>, and <math>-l<x<r</math> so that the body surface is entirely in the
finite region.

=== Solution in the finite region ===

We use the [[:Category:Boundary Element Method|Boundary Element Method]]
in the finite region.

===Solution in the Semi-infinite Domains===

We now solve Laplace's equation in the semi-infinite domains. First consider
the domain on the left so that <math>\Omega = \left\{ x<-l,\;-h\leq z\leq 0\right\} </math>
The equations are
{{standard linear wave scattering equations without body condition}}
We have the following explicit boundary conditions
<center><math>
\phi =f\left( z\right) ,\;\;x=-\,l
</math></center>
and
<center><math>
\lim\limits_{x\rightarrow -\infty }\phi \left( x,z\right) = \phi_0
e^{-k_{0}x}
+R\phi_0
e^{k_{0}x},
</math></center>
where <math>f\left( z\right) </math> is
an arbitrary continuous function<math>.</math> Our aim is to find the outward normal
derivative of the potential on <math>x=-l</math> as a function of <math>\tilde{\phi}\left(
z\right) </math>. <math>\phi_0</math> and <math>k_0</math> will be defined shortly
when we separate variables, but are equivalent to the [[Sommerfeld Radiation Condition]].

Since the water depth is constant in these regions we can solve Laplace's
equation by separation of variables.

{{separation of variables for a free surface}}

=== Expansion of the potential ===

This gives us the following expression for the potential in the region <math>
\Omega </math>
<center><math>
\phi \left( x,z\right) = \phi_0(z)e^{-k_0 x}
+\sum_{m=0}^{\infty } a_m \phi _{m}\left( z\right)
e^{k_{m}\left( x+l\right) }
</math></center>
where <math>a_m</math> can be found from <math>\left. \phi \right| _{x=-l} = f(z)</math>
from the orthogonality of the vertical eigenfunctions. Therefore
<center><math>
a_m +\delta_{m0}e^{k_0 l} = \frac{1}{A_m} \int_{-h}^{0} f(z) \phi_m(z) \mathrm{d}z
</math></center>

The normal derivative of potential (with the normal outward from the interior region) we obtain
<center><math>
\partial_n\left. \phi \right| _{x=-l}
= -\partial_x \left. \phi \right| _{x=-l}
= k_0 \phi_0(z)e^{k_0 l}
- \sum_{m=0}^{\infty } k_m a_m \phi _{m}\left( z\right)
</math></center>

If we define <math>\mathbf{Q}</math> as
<center><math>
\mathbf{Q} \left[f\left( z\right) \right]=
-\sum_{m=0}^{\infty } \frac{k_{m}}{A_m} \left\{ \int_{-h}^{0} f\left( s\right) \phi
_{m}\left( s\right) \mathrm{d}s \right\} \phi _{m}\left( z\right)
</math></center>
we can write
<center><math>
\partial_n \left. \phi \right| _{x=-l} =\mathbf{Q} \left[ f(z) \right] + 2k_0 \phi_0e^{k_0l},
</math></center>

Similarly, we now consider the potential in the region
<math>\Omega = \left\{ x>r,\;-h\leq z\leq 0\right\} </math>
which satisfies exactly the equations as before except the boundary
condition is
<center><math>
\lim\limits_{x\rightarrow \infty }\phi \left( x,z\right) = \phi_0
T e^{-k_{0}x}
</math></center>
We solve again by separation of variables and obtain
<center><math>
\left. \partial_n \phi \right| _{x=r}=
\left. \partial_x \phi \right| _{x=r}=\mathbf{Q}\left[f\left( z\right)\right] ,
</math></center>
where the outward normal is with respect to the inner domain as before. Note that, if
the depth is different on each side then the matrix <math>\mathbf{Q}</math> will be different
(since we need to use a different depth solving the [[Dispersion Relation for a Free Surface]] etc.)

== Boundary Element Method ==

We have reduced the problem to Laplace's equation in a finite domain subject
to certain boundary conditions. These boundary
conditions give the outward normal derivative of the potential as a function
of the potential but this is not always a point-wise condition; on some
boundaries it is given by an integral equation. We must solve both Laplace's
equation and the integral equations numerically. We will solve Laplace's
equation by the [[:Category:Boundary Element Method|Boundary Element Method]] and the integral equations by
numerical integration. However, the same discretisation of the boundary will
be used for both numerical solutions.
We solve Laplace's equation by a modified constant panel method
which reduces it to the following matrix equation
<center><math>
\frac{1}{2}\vec{\phi}=\partial_n\mathbf{G}\vec{\phi}-\mathbf{G}\partial_n\vec{\phi}.
</math></center>
where <math>\vec{\phi}\mathcal{\ }</math>and <math>\partial_n\vec{\phi}</math>
are vectors which approximate the potential and its normal derivative
around the boundary <math>\partial \Omega </math>, and <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>
are matrices corresponding to the Green function and the outward normal
derivative of the Green function respectively. The method used to calculate
the elements of the matrices <math>\mathbf{G}</math> and <math>\partial_n\mathbf{G}</math> can be
found in [[:Category:Boundary Element Method|Boundary Element Method]].

The outward normal derivative of the potential, <math>\partial_n\vec{\phi},</math> and the
potential, <math>\vec{\phi},</math> are related by the conditions on the boundary <math>
\partial \Omega </math>. This can be expressed as
<center><math>
\partial_n\vec{\phi}=\mathbf{A}\,\vec{\phi}-\vec{f},
</math></center>
where <math>\mathbf{A}</math> is the block diagonal matrix given by
<center><math>
\mathbf{A}\mathbb{=}\left[
\begin{matrix}
\mathbf{Q} & & & \\
& \mathbf{Q} & & \\
& & \alpha \,\mathbf{I} & \\
& & & 0
\end{matrix}
\right] ,
</math></center>
where we have separated the boundaries to the semi-infinite domains, the free surface and the
sea floor and body boundary.
<math>\mathbf{Q}</math> is a matrix
approximation of the integral operator of the same name and <math>\vec{f}</math> is
the vector
<center><math>
\vec{f}=\left[
\begin{matrix}
-2k_{0}\phi_0 \,e^{k_{0}l } \\
0 \\
\vdots \\
0
\end{matrix}
\right] .
</math></center>

We obtain the following matrix equation for the
potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
which can be solved straightforwardly. The reflection and transmission
coefficients are calculated from the solution.

===Numerical Calculation of <math>\mathbf{Q}</math> ===

We begin by truncating to a finite number (<math>N</math>) of
evanescent modes,
<center><math>
\mathbf{Q}\left[ f \right] =-\sum_{m=0}^{N}k_{m}\int_{-h}^{0}
f\left( s\right) \phi_{m}\left( s\right) \mathrm{d}s
\frac{\phi _{m} \left( z\right)}{A_m} .
</math></center>
We calculate the integral
with the same panels as we used to approximate the integrals of the Green
function and its normal derivative . Similarly, we
assume that <math>f(s) \,</math> is constant over each panel and integrate <math>\phi
_{m}\left( s\right) </math> exactly. This gives us the
following matrix factorisation of <math>\mathbf{Q}</math>
<center><math>
\mathbf{Q}[f]=\mathbf{S}\,\mathbf{R}\,[f].
</math></center>
The components of the matrices <math>\mathbf{S}</math> and <math>\mathbf{R}</math> are
<center><math>
s_{jm} = -k_m\phi _{m}\left( z_{j}\right) ,
</math></center>
<center><math>
r_{mj} = \frac{1}{A_m} \int_{z_{j}-\Delta x/2}^{z_{j}+\Delta x/2}\phi
_{m}\left( s\right) \mathrm{d}s
</math></center>

=== Reflection and Transmission Coefficients ===

If we multiply the potential at the left (after subtracting the incident wave) and the right by
<math>\mathbf{R}</math> we can calculate the coefficients in the eigenfunction expansion, and
hence determine the reflection and transmission coefficient.
where <math>z_{j}</math> is the value of the <math>z</math> coordinate in the centre of the panel
and <math>\Delta x</math> is the panel length.

=== Singularity Expansion Method ===
The matrix equation for the potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
can be rewritten as
<center><math>
\mathbf{M}(k) \vec{x}=\vec{F}(k)
</math></center>
where <math>\vec{x}</math> represents the unknown potential values in the new notation. Resonant frequencies (or scattering frequencies) correspond to zero eigenvalues of the matrix <math>\mathbf{M}(k)</math> for a set of discrete points <math>k=k_p</math> in the upper half of the complex plane (due to the time dependence). Since
<center><math>
\vec{x}=\left( \mathbf{M}(k)\right)^{-1}\vec{F}(k)
</math></center>
then <math>\vec{x}</math> will be singular for these scattering frequencies. Each scattering frequency, corresponding to a zero eigenvalue of <math>\mathbf{M}(k)</math> will possess a corresponding eigenvector <math>\mathbf{u}_{k_p}<\math> satisfying
<center><math>
\mathbf{M}(k_p)\mathbf{u}_{k_p}=0
</math></center>
and also a related conjugate transpose eigenvector <math>\mathbf{u}^{*}_{k_p}<\math>
satisfying
<center><math>
\mathbf{M}(k_p)\mathbf{u}_{k_p}=0
</math></center>

== Matlab Code ==

{{fixed body 2d bem code}}

=== Additional code ===

This program requires
* {{free surface dispersion equation code}}
* {{boundary element code}}

[[Category:Boundary Element Method]]
[[Category:Eigenfunction Matching Method]]

Boundary Element Method for a Fixed Body in Finite Depth

2009-11-24T04:11:53Z

Cfitzgerald: /* Boundary Element Method */

{{complete pages}}

== Introduction ==

We show how to solve for the wave scattering by a rigid body in constant [[Finite Depth]] using
the [[:Category:Boundary Element Method|Boundary Element Method]]. The method can be modified
to account for a body which can move and this is described in
[[Boundary Element Method for a Floating Body in Finite Depth]]

== Equations ==

{{boundary value problem for a fixed body}}

{{incident plane wave}}

{{sommerfeld radiation condition two dimensions}}

== Solution Method ==

We divide the domain into three regions, <math>x<-l</math>,
<math>x>r</math>, and <math>-l<x<r</math> so that the body surface is entirely in the
finite region.

=== Solution in the finite region ===

We use the [[:Category:Boundary Element Method|Boundary Element Method]]
in the finite region.

===Solution in the Semi-infinite Domains===

We now solve Laplace's equation in the semi-infinite domains. First consider
the domain on the left so that <math>\Omega = \left\{ x<-l,\;-h\leq z\leq 0\right\} </math>
The equations are
{{standard linear wave scattering equations without body condition}}
We have the following explicit boundary conditions
<center><math>
\phi =f\left( z\right) ,\;\;x=-\,l
</math></center>
and
<center><math>
\lim\limits_{x\rightarrow -\infty }\phi \left( x,z\right) = \phi_0
e^{-k_{0}x}
+R\phi_0
e^{k_{0}x},
</math></center>
where <math>f\left( z\right) </math> is
an arbitrary continuous function<math>.</math> Our aim is to find the outward normal
derivative of the potential on <math>x=-l</math> as a function of <math>\tilde{\phi}\left(
z\right) </math>. <math>\phi_0</math> and <math>k_0</math> will be defined shortly
when we separate variables, but are equivalent to the [[Sommerfeld Radiation Condition]].

Since the water depth is constant in these regions we can solve Laplace's
equation by separation of variables.

{{separation of variables for a free surface}}

=== Expansion of the potential ===

This gives us the following expression for the potential in the region <math>
\Omega </math>
<center><math>
\phi \left( x,z\right) = \phi_0(z)e^{-k_0 x}
+\sum_{m=0}^{\infty } a_m \phi _{m}\left( z\right)
e^{k_{m}\left( x+l\right) }
</math></center>
where <math>a_m</math> can be found from <math>\left. \phi \right| _{x=-l} = f(z)</math>
from the orthogonality of the vertical eigenfunctions. Therefore
<center><math>
a_m +\delta_{m0}e^{k_0 l} = \frac{1}{A_m} \int_{-h}^{0} f(z) \phi_m(z) \mathrm{d}z
</math></center>

The normal derivative of potential (with the normal outward from the interior region) we obtain
<center><math>
\partial_n\left. \phi \right| _{x=-l}
= -\partial_x \left. \phi \right| _{x=-l}
= k_0 \phi_0(z)e^{k_0 l}
- \sum_{m=0}^{\infty } k_m a_m \phi _{m}\left( z\right)
</math></center>

If we define <math>\mathbf{Q}</math> as
<center><math>
\mathbf{Q} \left[f\left( z\right) \right]=
-\sum_{m=0}^{\infty } \frac{k_{m}}{A_m} \left\{ \int_{-h}^{0} f\left( s\right) \phi
_{m}\left( s\right) \mathrm{d}s \right\} \phi _{m}\left( z\right)
</math></center>
we can write
<center><math>
\partial_n \left. \phi \right| _{x=-l} =\mathbf{Q} \left[ f(z) \right] + 2k_0 \phi_0e^{k_0l},
</math></center>

Similarly, we now consider the potential in the region
<math>\Omega = \left\{ x>r,\;-h\leq z\leq 0\right\} </math>
which satisfies exactly the equations as before except the boundary
condition is
<center><math>
\lim\limits_{x\rightarrow \infty }\phi \left( x,z\right) = \phi_0
T e^{-k_{0}x}
</math></center>
We solve again by separation of variables and obtain
<center><math>
\left. \partial_n \phi \right| _{x=r}=
\left. \partial_x \phi \right| _{x=r}=\mathbf{Q}\left[f\left( z\right)\right] ,
</math></center>
where the outward normal is with respect to the inner domain as before. Note that, if
the depth is different on each side then the matrix <math>\mathbf{Q}</math> will be different
(since we need to use a different depth solving the [[Dispersion Relation for a Free Surface]] etc.)

== Boundary Element Method ==

We have reduced the problem to Laplace's equation in a finite domain subject
to certain boundary conditions. These boundary
conditions give the outward normal derivative of the potential as a function
of the potential but this is not always a point-wise condition; on some
boundaries it is given by an integral equation. We must solve both Laplace's
equation and the integral equations numerically. We will solve Laplace's
equation by the [[:Category:Boundary Element Method|Boundary Element Method]] and the integral equations by
numerical integration. However, the same discretisation of the boundary will
be used for both numerical solutions.
We solve Laplace's equation by a modified constant panel method
which reduces it to the following matrix equation
<center><math>
\frac{1}{2}\vec{\phi}=\partial_n\mathbf{G}\vec{\phi}-\mathbf{G}\partial_n\vec{\phi}.
</math></center>
where <math>\vec{\phi}\mathcal{\ }</math>and <math>\partial_n\vec{\phi}</math>
are vectors which approximate the potential and its normal derivative
around the boundary <math>\partial \Omega </math>, and <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>
are matrices corresponding to the Green function and the outward normal
derivative of the Green function respectively. The method used to calculate
the elements of the matrices <math>\mathbf{G}</math> and <math>\partial_n\mathbf{G}</math> can be
found in [[:Category:Boundary Element Method|Boundary Element Method]].

The outward normal derivative of the potential, <math>\partial_n\vec{\phi},</math> and the
potential, <math>\vec{\phi},</math> are related by the conditions on the boundary <math>
\partial \Omega </math>. This can be expressed as
<center><math>
\partial_n\vec{\phi}=\mathbf{A}\,\vec{\phi}-\vec{f},
</math></center>
where <math>\mathbf{A}</math> is the block diagonal matrix given by
<center><math>
\mathbf{A}\mathbb{=}\left[
\begin{matrix}
\mathbf{Q} & & & \\
& \mathbf{Q} & & \\
& & \alpha \,\mathbf{I} & \\
& & & 0
\end{matrix}
\right] ,
</math></center>
where we have separated the boundaries to the semi-infinite domains, the free surface and the
sea floor and body boundary.
<math>\mathbf{Q}</math> is a matrix
approximation of the integral operator of the same name and <math>\vec{f}</math> is
the vector
<center><math>
\vec{f}=\left[
\begin{matrix}
-2k_{0}\phi_0 \,e^{k_{0}l } \\
0 \\
\vdots \\
0
\end{matrix}
\right] .
</math></center>

We obtain the following matrix equation for the
potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
which can be solved straightforwardly. The reflection and transmission
coefficients are calculated from the solution.

===Numerical Calculation of <math>\mathbf{Q}</math> ===

We begin by truncating to a finite number (<math>N</math>) of
evanescent modes,
<center><math>
\mathbf{Q}\left[ f \right] =-\sum_{m=0}^{N}k_{m}\int_{-h}^{0}
f\left( s\right) \phi_{m}\left( s\right) \mathrm{d}s
\frac{\phi _{m} \left( z\right)}{A_m} .
</math></center>
We calculate the integral
with the same panels as we used to approximate the integrals of the Green
function and its normal derivative . Similarly, we
assume that <math>f(s) \,</math> is constant over each panel and integrate <math>\phi
_{m}\left( s\right) </math> exactly. This gives us the
following matrix factorisation of <math>\mathbf{Q}</math>
<center><math>
\mathbf{Q}[f]=\mathbf{S}\,\mathbf{R}\,[f].
</math></center>
The components of the matrices <math>\mathbf{S}</math> and <math>\mathbf{R}</math> are
<center><math>
s_{jm} = -k_m\phi _{m}\left( z_{j}\right) ,
</math></center>
<center><math>
r_{mj} = \frac{1}{A_m} \int_{z_{j}-\Delta x/2}^{z_{j}+\Delta x/2}\phi
_{m}\left( s\right) \mathrm{d}s
</math></center>

=== Reflection and Transmission Coefficients ===

If we multiply the potential at the left (after subtracting the incident wave) and the right by
<math>\mathbf{R}</math> we can calculate the coefficients in the eigenfunction expansion, and
hence determine the reflection and transmission coefficient.
where <math>z_{j}</math> is the value of the <math>z</math> coordinate in the centre of the panel
and <math>\Delta x</math> is the panel length.

=== Singularity Expansion Method ===
The matrix equation for the potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
can be rewritten as
<center><math>
\left( \mathbf{M}(k)\right) \vec{x}=\vec{F}(k)
</math></center>
where <math>\vec{x}</math> represents the unknown potential values in the new notation. Resonant frequencies (or scattering frequencies) correspond to zero eigenvalues of the matrix <math>\mathbf{M}(k)</math> for a set of discrete points <math>k=k_p</math> in the upper half of the complex plane (due to the time dependence).

== Matlab Code ==

{{fixed body 2d bem code}}

=== Additional code ===

This program requires
* {{free surface dispersion equation code}}
* {{boundary element code}}

[[Category:Boundary Element Method]]
[[Category:Eigenfunction Matching Method]]

Boundary Element Method for a Fixed Body in Finite Depth

2009-11-24T04:06:22Z

Cfitzgerald: /* Boundary Element Method */

{{complete pages}}

== Introduction ==

We show how to solve for the wave scattering by a rigid body in constant [[Finite Depth]] using
the [[:Category:Boundary Element Method|Boundary Element Method]]. The method can be modified
to account for a body which can move and this is described in
[[Boundary Element Method for a Floating Body in Finite Depth]]

== Equations ==

{{boundary value problem for a fixed body}}

{{incident plane wave}}

{{sommerfeld radiation condition two dimensions}}

== Solution Method ==

We divide the domain into three regions, <math>x<-l</math>,
<math>x>r</math>, and <math>-l<x<r</math> so that the body surface is entirely in the
finite region.

=== Solution in the finite region ===

We use the [[:Category:Boundary Element Method|Boundary Element Method]]
in the finite region.

===Solution in the Semi-infinite Domains===

We now solve Laplace's equation in the semi-infinite domains. First consider
the domain on the left so that <math>\Omega = \left\{ x<-l,\;-h\leq z\leq 0\right\} </math>
The equations are
{{standard linear wave scattering equations without body condition}}
We have the following explicit boundary conditions
<center><math>
\phi =f\left( z\right) ,\;\;x=-\,l
</math></center>
and
<center><math>
\lim\limits_{x\rightarrow -\infty }\phi \left( x,z\right) = \phi_0
e^{-k_{0}x}
+R\phi_0
e^{k_{0}x},
</math></center>
where <math>f\left( z\right) </math> is
an arbitrary continuous function<math>.</math> Our aim is to find the outward normal
derivative of the potential on <math>x=-l</math> as a function of <math>\tilde{\phi}\left(
z\right) </math>. <math>\phi_0</math> and <math>k_0</math> will be defined shortly
when we separate variables, but are equivalent to the [[Sommerfeld Radiation Condition]].

Since the water depth is constant in these regions we can solve Laplace's
equation by separation of variables.

{{separation of variables for a free surface}}

=== Expansion of the potential ===

This gives us the following expression for the potential in the region <math>
\Omega </math>
<center><math>
\phi \left( x,z\right) = \phi_0(z)e^{-k_0 x}
+\sum_{m=0}^{\infty } a_m \phi _{m}\left( z\right)
e^{k_{m}\left( x+l\right) }
</math></center>
where <math>a_m</math> can be found from <math>\left. \phi \right| _{x=-l} = f(z)</math>
from the orthogonality of the vertical eigenfunctions. Therefore
<center><math>
a_m +\delta_{m0}e^{k_0 l} = \frac{1}{A_m} \int_{-h}^{0} f(z) \phi_m(z) \mathrm{d}z
</math></center>

The normal derivative of potential (with the normal outward from the interior region) we obtain
<center><math>
\partial_n\left. \phi \right| _{x=-l}
= -\partial_x \left. \phi \right| _{x=-l}
= k_0 \phi_0(z)e^{k_0 l}
- \sum_{m=0}^{\infty } k_m a_m \phi _{m}\left( z\right)
</math></center>

If we define <math>\mathbf{Q}</math> as
<center><math>
\mathbf{Q} \left[f\left( z\right) \right]=
-\sum_{m=0}^{\infty } \frac{k_{m}}{A_m} \left\{ \int_{-h}^{0} f\left( s\right) \phi
_{m}\left( s\right) \mathrm{d}s \right\} \phi _{m}\left( z\right)
</math></center>
we can write
<center><math>
\partial_n \left. \phi \right| _{x=-l} =\mathbf{Q} \left[ f(z) \right] + 2k_0 \phi_0e^{k_0l},
</math></center>

Similarly, we now consider the potential in the region
<math>\Omega = \left\{ x>r,\;-h\leq z\leq 0\right\} </math>
which satisfies exactly the equations as before except the boundary
condition is
<center><math>
\lim\limits_{x\rightarrow \infty }\phi \left( x,z\right) = \phi_0
T e^{-k_{0}x}
</math></center>
We solve again by separation of variables and obtain
<center><math>
\left. \partial_n \phi \right| _{x=r}=
\left. \partial_x \phi \right| _{x=r}=\mathbf{Q}\left[f\left( z\right)\right] ,
</math></center>
where the outward normal is with respect to the inner domain as before. Note that, if
the depth is different on each side then the matrix <math>\mathbf{Q}</math> will be different
(since we need to use a different depth solving the [[Dispersion Relation for a Free Surface]] etc.)

== Boundary Element Method ==

We have reduced the problem to Laplace's equation in a finite domain subject
to certain boundary conditions. These boundary
conditions give the outward normal derivative of the potential as a function
of the potential but this is not always a point-wise condition; on some
boundaries it is given by an integral equation. We must solve both Laplace's
equation and the integral equations numerically. We will solve Laplace's
equation by the [[:Category:Boundary Element Method|Boundary Element Method]] and the integral equations by
numerical integration. However, the same discretisation of the boundary will
be used for both numerical solutions.
We solve Laplace's equation by a modified constant panel method
which reduces it to the following matrix equation
<center><math>
\frac{1}{2}\vec{\phi}=\partial_n\mathbf{G}\vec{\phi}-\mathbf{G}\partial_n\vec{\phi}.
</math></center>
where <math>\vec{\phi}\mathcal{\ }</math>and <math>\partial_n\vec{\phi}</math>
are vectors which approximate the potential and its normal derivative
around the boundary <math>\partial \Omega </math>, and <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>
are matrices corresponding to the Green function and the outward normal
derivative of the Green function respectively. The method used to calculate
the elements of the matrices <math>\mathbf{G}</math> and <math>\partial_n\mathbf{G}</math> can be
found in [[:Category:Boundary Element Method|Boundary Element Method]].

The outward normal derivative of the potential, <math>\partial_n\vec{\phi},</math> and the
potential, <math>\vec{\phi},</math> are related by the conditions on the boundary <math>
\partial \Omega </math>. This can be expressed as
<center><math>
\partial_n\vec{\phi}=\mathbf{A}\,\vec{\phi}-\vec{f},
</math></center>
where <math>\mathbf{A}</math> is the block diagonal matrix given by
<center><math>
\mathbf{A}\mathbb{=}\left[
\begin{matrix}
\mathbf{Q} & & & \\
& \mathbf{Q} & & \\
& & \alpha \,\mathbf{I} & \\
& & & 0
\end{matrix}
\right] ,
</math></center>
where we have separated the boundaries to the semi-infinite domains, the free surface and the
sea floor and body boundary.
<math>\mathbf{Q}</math> is a matrix
approximation of the integral operator of the same name and <math>\vec{f}</math> is
the vector
<center><math>
\vec{f}=\left[
\begin{matrix}
-2k_{0}\phi_0 \,e^{k_{0}l } \\
0 \\
\vdots \\
0
\end{matrix}
\right] .
</math></center>

We obtain the following matrix equation for the
potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
which can be solved straightforwardly. The reflection and transmission
coefficients are calculated from the solution.

===Numerical Calculation of <math>\mathbf{Q}</math> ===

We begin by truncating to a finite number (<math>N</math>) of
evanescent modes,
<center><math>
\mathbf{Q}\left[ f \right] =-\sum_{m=0}^{N}k_{m}\int_{-h}^{0}
f\left( s\right) \phi_{m}\left( s\right) \mathrm{d}s
\frac{\phi _{m} \left( z\right)}{A_m} .
</math></center>
We calculate the integral
with the same panels as we used to approximate the integrals of the Green
function and its normal derivative . Similarly, we
assume that <math>f(s) \,</math> is constant over each panel and integrate <math>\phi
_{m}\left( s\right) </math> exactly. This gives us the
following matrix factorisation of <math>\mathbf{Q}</math>
<center><math>
\mathbf{Q}[f]=\mathbf{S}\,\mathbf{R}\,[f].
</math></center>
The components of the matrices <math>\mathbf{S}</math> and <math>\mathbf{R}</math> are
<center><math>
s_{jm} = -k_m\phi _{m}\left( z_{j}\right) ,
</math></center>
<center><math>
r_{mj} = \frac{1}{A_m} \int_{z_{j}-\Delta x/2}^{z_{j}+\Delta x/2}\phi
_{m}\left( s\right) \mathrm{d}s
</math></center>

=== Reflection and Transmission Coefficients ===

If we multiply the potential at the left (after subtracting the incident wave) and the right by
<math>\mathbf{R}</math> we can calculate the coefficients in the eigenfunction expansion, and
hence determine the reflection and transmission coefficient.
where <math>z_{j}</math> is the value of the <math>z</math> coordinate in the centre of the panel
and <math>\Delta x</math> is the panel length.

=== Singularity Expansion Method ===
The matrix equation for the potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
can be rewritten as
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>

== Matlab Code ==

{{fixed body 2d bem code}}

=== Additional code ===

This program requires
* {{free surface dispersion equation code}}
* {{boundary element code}}

[[Category:Boundary Element Method]]
[[Category:Eigenfunction Matching Method]]

Boundary Element Method for a Fixed Body in Finite Depth

2009-11-24T04:04:04Z

Cfitzgerald: /* Reflection and Transmission Coefficients */

{{complete pages}}

== Introduction ==

We show how to solve for the wave scattering by a rigid body in constant [[Finite Depth]] using
the [[:Category:Boundary Element Method|Boundary Element Method]]. The method can be modified
to account for a body which can move and this is described in
[[Boundary Element Method for a Floating Body in Finite Depth]]

== Equations ==

{{boundary value problem for a fixed body}}

{{incident plane wave}}

{{sommerfeld radiation condition two dimensions}}

== Solution Method ==

We divide the domain into three regions, <math>x<-l</math>,
<math>x>r</math>, and <math>-l<x<r</math> so that the body surface is entirely in the
finite region.

=== Solution in the finite region ===

We use the [[:Category:Boundary Element Method|Boundary Element Method]]
in the finite region.

===Solution in the Semi-infinite Domains===

We now solve Laplace's equation in the semi-infinite domains. First consider
the domain on the left so that <math>\Omega = \left\{ x<-l,\;-h\leq z\leq 0\right\} </math>
The equations are
{{standard linear wave scattering equations without body condition}}
We have the following explicit boundary conditions
<center><math>
\phi =f\left( z\right) ,\;\;x=-\,l
</math></center>
and
<center><math>
\lim\limits_{x\rightarrow -\infty }\phi \left( x,z\right) = \phi_0
e^{-k_{0}x}
+R\phi_0
e^{k_{0}x},
</math></center>
where <math>f\left( z\right) </math> is
an arbitrary continuous function<math>.</math> Our aim is to find the outward normal
derivative of the potential on <math>x=-l</math> as a function of <math>\tilde{\phi}\left(
z\right) </math>. <math>\phi_0</math> and <math>k_0</math> will be defined shortly
when we separate variables, but are equivalent to the [[Sommerfeld Radiation Condition]].

Since the water depth is constant in these regions we can solve Laplace's
equation by separation of variables.

{{separation of variables for a free surface}}

=== Expansion of the potential ===

This gives us the following expression for the potential in the region <math>
\Omega </math>
<center><math>
\phi \left( x,z\right) = \phi_0(z)e^{-k_0 x}
+\sum_{m=0}^{\infty } a_m \phi _{m}\left( z\right)
e^{k_{m}\left( x+l\right) }
</math></center>
where <math>a_m</math> can be found from <math>\left. \phi \right| _{x=-l} = f(z)</math>
from the orthogonality of the vertical eigenfunctions. Therefore
<center><math>
a_m +\delta_{m0}e^{k_0 l} = \frac{1}{A_m} \int_{-h}^{0} f(z) \phi_m(z) \mathrm{d}z
</math></center>

The normal derivative of potential (with the normal outward from the interior region) we obtain
<center><math>
\partial_n\left. \phi \right| _{x=-l}
= -\partial_x \left. \phi \right| _{x=-l}
= k_0 \phi_0(z)e^{k_0 l}
- \sum_{m=0}^{\infty } k_m a_m \phi _{m}\left( z\right)
</math></center>

If we define <math>\mathbf{Q}</math> as
<center><math>
\mathbf{Q} \left[f\left( z\right) \right]=
-\sum_{m=0}^{\infty } \frac{k_{m}}{A_m} \left\{ \int_{-h}^{0} f\left( s\right) \phi
_{m}\left( s\right) \mathrm{d}s \right\} \phi _{m}\left( z\right)
</math></center>
we can write
<center><math>
\partial_n \left. \phi \right| _{x=-l} =\mathbf{Q} \left[ f(z) \right] + 2k_0 \phi_0e^{k_0l},
</math></center>

Similarly, we now consider the potential in the region
<math>\Omega = \left\{ x>r,\;-h\leq z\leq 0\right\} </math>
which satisfies exactly the equations as before except the boundary
condition is
<center><math>
\lim\limits_{x\rightarrow \infty }\phi \left( x,z\right) = \phi_0
T e^{-k_{0}x}
</math></center>
We solve again by separation of variables and obtain
<center><math>
\left. \partial_n \phi \right| _{x=r}=
\left. \partial_x \phi \right| _{x=r}=\mathbf{Q}\left[f\left( z\right)\right] ,
</math></center>
where the outward normal is with respect to the inner domain as before. Note that, if
the depth is different on each side then the matrix <math>\mathbf{Q}</math> will be different
(since we need to use a different depth solving the [[Dispersion Relation for a Free Surface]] etc.)

== Boundary Element Method ==

We have reduced the problem to Laplace's equation in a finite domain subject
to certain boundary conditions. These boundary
conditions give the outward normal derivative of the potential as a function
of the potential but this is not always a point-wise condition; on some
boundaries it is given by an integral equation. We must solve both Laplace's
equation and the integral equations numerically. We will solve Laplace's
equation by the [[:Category:Boundary Element Method|Boundary Element Method]] and the integral equations by
numerical integration. However, the same discretisation of the boundary will
be used for both numerical solutions.
We solve Laplace's equation by a modified constant panel method
which reduces it to the following matrix equation
<center><math>
\frac{1}{2}\vec{\phi}=\partial_n\mathbf{G}\vec{\phi}-\mathbf{G}\partial_n\vec{\phi}.
</math></center>
where <math>\vec{\phi}\mathcal{\ }</math>and <math>\partial_n\vec{\phi}</math>
are vectors which approximate the potential and its normal derivative
around the boundary <math>\partial \Omega </math>, and <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>
are matrices corresponding to the Green function and the outward normal
derivative of the Green function respectively. The method used to calculate
the elements of the matrices <math>\mathbf{G}</math> and <math>\partial_n\mathbf{G}</math> can be
found in [[:Category:Boundary Element Method|Boundary Element Method]].

The outward normal derivative of the potential, <math>\partial_n\vec{\phi},</math> and the
potential, <math>\vec{\phi},</math> are related by the conditions on the boundary <math>
\partial \Omega </math>. This can be expressed as
<center><math>
\partial_n\vec{\phi}=\mathbf{A}\,\vec{\phi}-\vec{f},
</math></center>
where <math>\mathbf{A}</math> is the block diagonal matrix given by
<center><math>
\mathbf{A}\mathbb{=}\left[
\begin{matrix}
\mathbf{Q} & & & \\
& \mathbf{Q} & & \\
& & \alpha \,\mathbf{I} & \\
& & & 0
\end{matrix}
\right] ,
</math></center>
where we have separated the boundaries to the semi-infinite domains, the free surface and the
sea floor and body boundary.
<math>\mathbf{Q}</math> is a matrix
approximation of the integral operator of the same name and <math>\vec{f}</math> is
the vector
<center><math>
\vec{f}=\left[
\begin{matrix}
-2k_{0}\phi_0 \,e^{k_{0}l } \\
0 \\
\vdots \\
0
\end{matrix}
\right] .
</math></center>

We obtain the following matrix equation for the
potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
which can be solved straightforwardly. The reflection and transmission
coefficients are calculated from the solution.

===Numerical Calculation of <math>\mathbf{Q}</math> ===

We begin by truncating to a finite number (<math>N</math>) of
evanescent modes,
<center><math>
\mathbf{Q}\left[ f \right] =-\sum_{m=0}^{N}k_{m}\int_{-h}^{0}
f\left( s\right) \phi_{m}\left( s\right) \mathrm{d}s
\frac{\phi _{m} \left( z\right)}{A_m} .
</math></center>
We calculate the integral
with the same panels as we used to approximate the integrals of the Green
function and its normal derivative . Similarly, we
assume that <math>f(s) \,</math> is constant over each panel and integrate <math>\phi
_{m}\left( s\right) </math> exactly. This gives us the
following matrix factorisation of <math>\mathbf{Q}</math>
<center><math>
\mathbf{Q}[f]=\mathbf{S}\,\mathbf{R}\,[f].
</math></center>
The components of the matrices <math>\mathbf{S}</math> and <math>\mathbf{R}</math> are
<center><math>
s_{jm} = -k_m\phi _{m}\left( z_{j}\right) ,
</math></center>
<center><math>
r_{mj} = \frac{1}{A_m} \int_{z_{j}-\Delta x/2}^{z_{j}+\Delta x/2}\phi
_{m}\left( s\right) \mathrm{d}s
</math></center>

=== Reflection and Transmission Coefficients ===

If we multiply the potential at the left (after subtracting the incident wave) and the right by
<math>\mathbf{R}</math> we can calculate the coefficients in the eigenfunction expansion, and
hence determine the reflection and transmission coefficient.
where <math>z_{j}</math> is the value of the <math>z</math> coordinate in the centre of the panel
and <math>\Delta x</math> is the panel length.

=== Singularity Expansion Method ===

== Matlab Code ==

{{fixed body 2d bem code}}

=== Additional code ===

This program requires
* {{free surface dispersion equation code}}
* {{boundary element code}}

[[Category:Boundary Element Method]]
[[Category:Eigenfunction Matching Method]]

Green Function Method for a Floating Body on the Surface

2009-09-24T00:31:04Z

Cfitzgerald: /* Equation in Terms of the Modes */

{{complete pages}}

== Introduction ==

The problem of a two-dimensional floating body which has negligible submergence
is solved using a green function.
The problem of a dock is solved in [[Green Function Method for a Finite Dock]]
and for a floating elastic plate is solved
in [[Green Function Methods for Floating Elastic Plates]]

== Equations for a Finite Plate in Frequency Domain ==

{{finite floating body on the surface frequency domain}}

== Equation in Terms of the Modes ==

The equations are
{{general dock type body equations}}
<center><math>
\mathrm{i}\omega\sum_{n=0,1}\zeta_{n}X_{n} =\partial_{z}\phi,\,\,x\in
(-L,L),\,\, z=0,
</math></center>
<center><math>
\left(- \alpha\gamma + 1\right) \zeta_{n}=-i\omega
\int_{-L}^{L}\phi X_{n}\mathrm{d}x, \,\,x\in
(-L,L),\,\, z=0,
</math></center>
We solve for the potential (and displacement) as the sum of
the diffracted and radiation potentials in the standard way,
as for a rigid body.
<center><math>
\phi=\phi^{\mathrm{D}}+\phi^{\mathrm{R}} ,\,
</math></center>
We begin with the diffraction potential <math>\phi^{\mathrm{D}}</math> which
satisfies the following equations
{{diffraction potential equations for a dock}}
{{radiation condition for diffracted potential}}

We now consider the radiation potentials <math>\phi^{\mathrm{R}}</math>. We can express the radiation potential as:
<center><math>
\phi^{\mathrm{R}}=\sum_{n=0,1}\zeta_n \phi_n^{\mathrm{R}}
</math></center>
which satisfy the following equations
{{radiation potential equations for a dock like structure}}
Therefore we find the potential as
<center><math>
\left( - \alpha\gamma + 1\right) \zeta_{n}=-i\omega
\int_{-L}^{L}\phi^{\mathrm{D}} X_{n}\mathrm{d}x +
\sum_{m=0,1}\left(\omega^2 a_{mn}(\omega) - i\omega b_{mn}(\omega)\right)
\zeta_{m},
</math></center>
where the functions <math>a_{mn}(\omega)</math> and <math>b_{mn}(\omega)</math> are given by
<center><math>
\omega^2 a_{mn}(\omega) -i\omega b_{mn}(\omega) = - i\omega\int_{-L}^{L}\phi_m^{\mathrm{R}}X_{n}\mathrm{d}x,
</math></center>
and they are referred to as the added mass and damping coefficients (see [[Added-Mass, Damping Coefficients And Exciting Forces]])
respectively.

Note that for this simple example the added mass and damping matrices are diagonal.

== Solution for the Radiation and Diffracted Potential ==

{{Green's function equations for the diffracted potential}}

and

{{Green's function equations for the radiation potential for a dock like structure}}

== Reflection and Transmission Coefficients ==

{{derivation of reflection and transmission in two dimensions}}

== Matlab Code ==

{{rigid plate modes code}}

Boundary Element Method for a Fixed Body in Finite Depth

2009-09-16T03:31:54Z

Cfitzgerald:

{{complete pages}}

== Introduction ==

We show how to solve for the wave scattering by a rigid body in constant [[Finite Depth]] using
the [[:Category:Boundary Element Method|Boundary Element Method]]. The method can be modified
to account for a body which can move and this is described in
[[Boundary Element Method for a Floating Body in Finite Depth]]

== Equations ==

{{boundary value problem for a fixed body}}

{{incident plane wave}}

{{sommerfeld radiation condition two dimensions}}

== Solution Method ==

We divide the domain into three regions, <math>x<-l</math>,
<math>x>r</math>, and <math>-l<x<r</math> so that the body surface is entirely in the
finite region.

=== Solution in the finite region ===

We use the [[:Category:Boundary Element Method|Boundary Element Method]]
in the finite region.

===Solution in the Semi-infinite Domains===

We now solve Laplace's equation in the semi-infinite domains. First consider
the domain on the left so that <math>\Omega = \left\{ x<-l,\;-h\leq z\leq 0\right\} </math>
The equations are
{{standard linear wave scattering equations without body condition}}
We have the following explicit boundary conditions
<center><math>
\phi =f\left( z\right) ,\;\;x=-\,l
</math></center>
and
<center><math>
\lim\limits_{x\rightarrow -\infty }\phi \left( x,z\right) = \phi_0
e^{-k_{0}x}
+R\phi_0
e^{k_{0}x},
</math></center>
where <math>f\left( z\right) </math> is
an arbitrary continuous function<math>.</math> Our aim is to find the outward normal
derivative of the potential on <math>x=-l</math> as a function of <math>\tilde{\phi}\left(
z\right) </math>. <math>\phi_0</math> and <math>k_0</math> will be defined shortly
when we separate variables, but are equivalent to the [[Sommerfeld Radiation Condition]].

Since the water depth is constant in these regions we can solve Laplace's
equation by separation of variables.

{{separation of variables for a free surface}}

=== Expansion of the potential ===

This gives us the following expression for the potential in the region <math>
\Omega </math>
<center><math>
\phi \left( x,z\right) = \phi_0(z)e^{-k_0 x}
+\sum_{m=0}^{\infty } a_m \phi _{m}\left( z\right)
e^{k_{m}\left( x+l\right) }
</math></center>
where <math>a_m</math> can be found from <math>\left. \phi \right| _{x=-l} = f(z)</math>
from the orthogonality of the vertical eigenfunctions. Therefore
<center><math>
a_m +\delta_{m0}e^{k_0 l} = \frac{1}{A_m} \int_{-h}^{0} f(z) \phi_m(z) \mathrm{d}z
</math></center>

The normal derivative of potential (with the normal outward from the interior region) we obtain
<center><math>
\partial_n\left. \phi \right| _{x=-l}
= -\partial_x \left. \phi \right| _{x=-l}
= k_0 \phi_0(z)e^{k_0 l}
- \sum_{m=0}^{\infty } k_m a_m \phi _{m}\left( z\right)
</math></center>

If we define <math>\mathbf{Q}</math> as
<center><math>
\mathbf{Q} \left[f\left( z\right) \right]=
-\sum_{m=0}^{\infty } \frac{k_{m}}{A_m} \left\{ \int_{-h}^{0} f\left( s\right) \phi
_{m}\left( s\right) \mathrm{d}s \right\} \phi _{m}\left( z\right)
</math></center>
we can write
<center><math>
\partial_n \left. \phi \right| _{x=-l} =\mathbf{Q} \left[ f(z) \right] + 2k_0 \phi_0e^{k_0l},
</math></center>

Similarly, we now consider the potential in the region
<math>\Omega = \left\{ x>r,\;-h\leq z\leq 0\right\} </math>
which satisfies exactly the equations as before except the boundary
condition is
<center><math>
\lim\limits_{x\rightarrow \infty }\phi \left( x,z\right) = \phi_0
T e^{-k_{0}x}
</math></center>
We solve again by separation of variables and obtain
<center><math>
\left. \partial_n \phi \right| _{x=r}=
\left. \partial_x \phi \right| _{x=r}=\mathbf{Q}\left[f\left( z\right)\right] ,
</math></center>
where the outward normal is with respect to the inner domain as before. Note that, if
the depth is different on each side then the matrix <math>\mathbf{Q}</math> will be different
(since we need to use a different depth solving the [[Dispersion Relation for a Free Surface]] etc.)

== Boundary Element Method ==

We have reduced the problem to Laplace's equation in a finite domain subject
to certain boundary conditions. These boundary
conditions give the outward normal derivative of the potential as a function
of the potential but this is not always a point-wise condition; on some
boundaries it is given by an integral equation. We must solve both Laplace's
equation and the integral equations numerically. We will solve Laplace's
equation by the [[:Category:Boundary Element Method|Boundary Element Method]] and the integral equations by
numerical integration. However, the same discretisation of the boundary will
be used for both numerical solutions.
We solve Laplace's equation by a modified constant panel method
which reduces it to the following matrix equation
<center><math>
\frac{1}{2}\vec{\phi}=\partial_n\mathbf{G}\vec{\phi}-\mathbf{G}\partial_n\vec{\phi}.
</math></center>
where <math>\vec{\phi}\mathcal{\ }</math>and <math>\partial_n\vec{\phi}</math>
are vectors which approximate the potential and its normal derivative
around the boundary <math>\partial \Omega </math>, and <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>
are matrices corresponding to the Green function and the outward normal
derivative of the Green function respectively. The method used to calculate
the elements of the matrices <math>\mathbf{G}</math> and <math>\partial_n\mathbf{G}</math> can be
found in [[:Category:Boundary Element Method|Boundary Element Method]].

The outward normal derivative of the potential, <math>\partial_n\vec{\phi},</math> and the
potential, <math>\vec{\phi},</math> are related by the conditions on the boundary <math>
\partial \Omega </math>. This can be expressed as
<center><math>
\partial_n\vec{\phi}=\mathbf{A}\,\vec{\phi}-\vec{f},
</math></center>
where <math>\mathbf{A}</math> is the block diagonal matrix given by
<center><math>
\mathbf{A}\mathbb{=}\left[
\begin{matrix}
\mathbf{Q} & & & \\
& \mathbf{Q} & & \\
& & \alpha \,\mathbf{I} & \\
& & & 0
\end{matrix}
\right] ,
</math></center>
where we have separated the boundaries to the semi-infinite domains, the free surface and the
sea floor and body boundary.
<math>\mathbf{Q}</math> is a matrix
approximation of the integral operator of the same name and <math>\vec{f}</math> is
the vector
<center><math>
\vec{f}=\left[
\begin{matrix}
-2k_{0}\phi_0 \,e^{k_{0}l } \\
0 \\
\vdots \\
0
\end{matrix}
\right] .
</math></center>

We obtain the following matrix equation for the
potential
<center><math>
\left( \frac{1}{2}\ \mathbf{I}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
which can be solved straightforwardly. The reflection and transmission
coefficients are calculated from the solution.

===Numerical Calculation of <math>\mathbf{Q}</math> ===

We begin by truncating to a finite number (<math>N</math>) of
evanescent modes,
<center><math>
\mathbf{Q}\left[ f \right] =-\sum_{m=0}^{N}k_{m}\int_{-h}^{0}
f\left( s\right) \phi_{m}\left( s\right) \mathrm{d}s
\frac{\phi _{m} \left( z\right)}{A_m} .
</math></center>
We calculate the integral
with the same panels as we used to approximate the integrals of the Green
function and its normal derivative . Similarly, we
assume that <math>f(s) \,</math> is constant over each panel and integrate <math>\phi
_{m}\left( s\right) </math> exactly. This gives us the
following matrix factorisation of <math>\mathbf{Q}</math>
<center><math>
\mathbf{Q}[f]=\mathbf{S}\,\mathbf{R}\,[f].
</math></center>
The components of the matrices <math>\mathbf{S}</math> and <math>\mathbf{R}</math> are
<center><math>
s_{jm} = -k_m\phi _{m}\left( z_{j}\right) ,
</math></center>
<center><math>
r_{mj} = \frac{1}{A_m} \int_{z_{j}-\Delta x/2}^{z_{j}+\Delta x/2}\phi
_{m}\left( s\right) \mathrm{d}s
</math></center>

=== Reflection and Transmission Coefficients ===

If we multiply the potential at the left (after subtracting the incident wave) and the right by
<math>\mathbf{R}</math> we can calculate the coefficients in the eigenfunction expansion, and
hence determine the reflection and transmission coefficient.
where <math>z_{j}</math> is the value of the <math>z</math> coordinate in the centre of the panel
and <math>\Delta x</math> is the panel length.

== Matlab Code ==

{{fixed body 2d bem code}}

=== Additional code ===

This program requires
* {{free surface dispersion equation code}}
* {{boundary element code}}

[[Category:Boundary Element Method]]
[[Category:Eigenfunction Matching Method]]

Category:Time-Dependent Linear Water Waves

2009-09-03T01:57:06Z

Cfitzgerald: /* Floating body constrained to move in heave */

[[Category:Linear Water-Wave Theory]]

{{incomplete pages}}

Generally the focus of research is on the [[Frequency Domain Problem]]. The time-domain
problem can be solved by [[Generalized Eigenfunction Expansion]] or by
an [[Expansion over the Resonances]] or using [[Memory Effect Function]].

== Introduction ==

Analytical solutions of the linearised time-domain equations for a coupled-motion or forced-motion problems are very rare. In fact, only a small number of analytic time-domain solutions have been obtained (see the papers by [[Kennard 1949]] and [[McIver 1994]]) for structures with simple geometries. Therefore, a variety of numerical methods have been developed to solve initial-value time-domain wave-structure interaction problems. The Fourier transform enables the time and frequency domain quantities to be related and the frequency domain forces and potentials have important roles in the development of the numerical time-domain solution methods. This can be partly attributed to the fact that, in the past, analytical or semi-analytical solutions to linearised water-wave problems were more easily obtained in the frequency domain given that the assumption of time-harmonic motion results in a significant simplification of the interaction equations. Therefore, it was natural to develop time-domain solutions based on frequency domain results and the Fourier transform relation between the domains.

== Equations of motion in the Time Domain ==

=== Two Dimensional Equations for fixed bodies in the time domain ===

{{Equations for fixed bodies in the time domain}}

=== Two dimensional equations for a floating body ===
We now consider the equations for a floating two-dimensional structure.
{{equations of motion time domain without body condition}}

The body boundary condition for a fixed body is given in terms
of the 3 rigid body motions, namely surge, heave and pitch which are indexed as <math>\mu=1,3,5</math> in order to be consistent with the three-dimensional problem. We have a kinematic condition
<center><math>
\partial_{n}\Phi=\sum\partial_t \xi_{\nu}\mathbf{n}_{\nu},\ \mathbf{x}\in\partial\Omega,
</math></center>
where <math>\xi_{\nu}</math> is the motion of the <math>\mu</math>th mode and
<math>\mathbf{n}_{\nu}</math> is the normal associated with this mode.
The dynamic condition is the equation of motion for the structure:
<center><math>
M_{\mu\nu}\partial_t^2 \xi_{\nu}=-\rho\iint_{\partial\Omega}\partial_t\Phi n_{\mu}\, dS- c_{\mu\nu}\xi_{\nu},\quad \textrm{for} \qquad \mu=1,3,5,
</math></center>
where summation over repeated indices is assumed. In this equation, <math>M_{\mu\nu}</math> are the elements of the mass matrix
<center><math>
\mathbf{M}=\left[
\begin{matrix}
M & 0 & M(z^c-Z^R) \\
0 & M & -M(x^c-X^R) \\
M(z^c-Z^R)& -M(x^c-X^R) & I^b_{11}+I^b_{33}
\end{matrix}
\right] ,
</math></center>
for the structure and <math>c_{\mu\nu}</math> are the elements of the buoyancy matrix
<center><math>
\mathbf{C}=\left[
\begin{matrix}
0 & 0 & 0 \\
0 & \rho g W & -\rho g I^A_{1} \\
0 & -\rho g I^A_{1} & \begin{matrix}\rho g (I^A_{11}+I^V_3)-\\
Mg(z^c-Z^R) \end{matrix}
\end{matrix}
\right].
</math></center>
The terms <math>I^b_{11}</math>, <math>I^b_{33}</math> are the moments of inertia of the body about the <math>x</math> and <math>z</math> axes and the terms <math>I_1^{A}</math>, <math>I^{A}_{11}</math> are the first and second moments of the waterplane (the waterplane area is denoted <math>W</math>) about the <math>x</math>-axis (see Chapter 7, [[Mei 1983]]). In addition, <math>(x^c,z^c)</math> and <math>(X^R,Z^R)</math> are the positions of the centre of mass and centre of rotation of the body and <math>I^{V}_{3}</math> is <math>z</math>-component centre of buoyancy of the structure. Thus, the coupled equations of motion for a floating structure have been derived. (N.B. if is assumed that the centre of rotation and the centre of mass of the structure coincide, i.e. if it is assumed that the body is semi-submerged, the mass and buoyancy matrices become diagonal). Any wave incidence is assumed to be propagating in the positive <math>x</math> direction.) The scattering and radiation problems are simpler than the coupled problem because the motion of the the structure is then prescribed.

The initial conditions are
{{initial free surface time domain}}
and
{{initial conditions floating body}}

=== Floating body constrained to move in heave ===
The simplest type of floating body problem concerns the motion of a body constrained to move in heave in two-dimensions. Apart from the boundary condition on the structure surface which becomes
<center><math>
\partial_n \Phi = \partial_t\xi_{3}\mathbf{n}_{3},
</math></center>
where <math>\xi_{3}</math> is the heave displacement of the structure, the equations governing the motion of the fluid remain the same. The equation of motion for a body constrained to move in heave only is
<center><math>
M\partial_t^2 \xi_{3}=-\rho\iint_{\partial\Omega}\partial_t\Phi n_{3}\, dS- \rho g W\xi_{3}.
</math></center>
which is a significant simplification compared to the general case - the mass matrix and buoyancy matrix are replaced simply by the mass <math>M</math> and the hydrostatic term <math>\rho g W</math>. The initial conditions for the fluid and the structure (<math>\xi_{3}(0)</math>, <math>\dot{\xi}_{3}(0)</math>) must also be prescribed to complete the problem specification.

==Frequency domain definitions==

Linearised water-wave problems were first investigated in the frequency domain wherein all motions are assumed to be time-harmonic and to have begun from <math>t= -\infty</math> . The time-dependence can be removed in a straightforward manner by substituting for the potential and any structural motions using
<center><math>
\Phi(\mathbf{x},t)=Re \{\phi(\mathbf{x},\omega) e^{-i\omega t}\}
</math></center>
<center><math>
V(\mathbf{x},t)=Re \{ v(\mathbf{x},\omega) e^{-i\omega t}\}
</math></center>
and this results in a considerable simplification of the problem. The complex amplitudes of the dynamic quantities vary with frequency and by removing the time-dependence the governing equations and boundary conditions will feature these terms only.
However, \citeasnoun{mciver2003} have shown that the frequency-domain potential can also be defined using the following operations on the time-domain solution:

*take a Laplace transform of the time-domain potential
<center><math>
\hat{\phi}(\mathbf{x},s)=\int^{\infty}_{0}\Phi(\mathbf{x},t)e^{-s t}\, dt ,\quad \textrm{Re } s >0;
</math></center>

*apply the change of variables <math>s=-i\omega</math> so that
<center><math>
\phi(\mathbf{x},\omega)=\hat{\phi}(\mathbf{x},-i\omega).
</math></center>

*noting that <math>\phi(\mathbf{x},-\omega)=\bar{\phi}(\mathbf{x},\omega)</math>, the inverse Fourier transform is given by
<center><math>(3)
\Phi(\mathbf{x},t)=\frac{1}{2\pi}\int^{\infty}_{-\infty}\phi(\mathbf{x},\omega)e^{-i\omega t} d\omega =\frac{1}{\pi}Re\int^{\infty}_{0}\phi(\mathbf{x},\omega)e^{-i\omega t}\,d\omega
</math></center>
where the path of integration must pass over any singularities of <math>\phi</math> that lie on the real axis and it has been assumed that there is no motion prior to <math>t=0</math>, i.e. <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math>.

If <math>\omega</math> is real then <math>\phi(\mathbf{x},\omega)</math> is just the frequency domain potential. By the principle of causality, which precludes the existence of the response before the cause so that <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math> , no singularities can exist in <math>\textrm{Im } \omega>0</math>. Therefore, if it is assumed that no singularities exist on the real axis the inverse transform will integrate the frequency-domain potential over the full range of <math>\omega</math>. With this approach, the frequency-domain equations are obtained by applying the Fourier transform to the time-dependent equations. However, unlike the equations resulting from the assumptions~(2) of time-harmonic motion from <math>t= -\infty</math>, the initial conditions will be present in the frequency-domain equations arising from the Fourier transformation. Although, the latter equations are more mathematically consistent it is usual to set all initial conditions to zero before attempting to solve the equations even if this is not the case. Nevertheless, in the frequency-domain analysis of resonances in coupled motion problems \citeasnoun{pmciver2005} retains the initial condition terms in the equation of motion for the structure and obtains some important results regarding the resonant behaviour of the motion. Therefore, the retention of the initial condition terms depends on the application of the subsequent equations.

Many analytical and approximate solutions exist for frequency-domain problems involving structures with simple geometries. Thus, in the simplest frequency-domain radiation and diffraction problems it is possible to completely determine the velocity potential analytically. The hydrodynamic forces, which are crucial in the context of marine engineering, can then be calculated directly from the potential. For more general geometries this is not possible and the hydrodynamic forces must be computed using numerical methods. Nevertheless, it is useful to consider the analytical expressions for the various potentials and forces in the frequency-domain problem.

In a linearised coupled motion problem, it is customary to decompose the total velocity potential into a scattering potential <math>\phi^{S}</math> and a radiation potential <math>\phi^{R}</math>. Thus, the coupled motion problem will require the solution of the radiation problem and the scattering problem. The radiation potential can be further decomposed into a sum over the six modes of motion so that the total frequency domain potential is
<center><math>
\phi=\phi^{S}+\sum_{\mu}v_{\mu}\phi_{\mu}
</math></center>

where <math>u_{\mu}</math> is the complex amplitude of the generalised velocity in the <math>\mu</math> direction and <math>\phi_{\mu}</math> describes the fluid response due to the forced oscillations in mode <math>\mu</math> with unit velocity amplitude. The velocities in each mode will be determined from the frequency domain equation of motion for the body. However, this requires the computation of the radiation and exciting forces on the structure. Therefore, it is necessary to consider the radiation problem prior to the coupled motion equation. It is first important to note that, for a given mode <math>\mu</math> the boundary condition on the structure will be

<center><math>

\frac{\partial\phi_{\mu}}{\partial n}=n_{\mu},

</math></center>

because the total velocity is

<center><math>

v(\omega)=\sum_{\mu}v_{\mu}n_{\mu}

</math></center>

where <math>n_{\mu}</math> is the <math>\mu</math> component of the generalised normal and <math>v_{\mu}</math> is the component of the generalised complex velocity amplitude in this direction. The hydrodynamic force on the structure in the <math>\mu</math> direction due to the fluid response to the forced oscillations is

<center><math>(5)

F_{\mu}^{R}=i\omega\rho\iint_{S_{B}}\phi^{R}n_{\mu} \,dS

</math></center>

and the force due to the diffraction of an incident wave by the fixed structure is

<center><math>(6)

F_{\mu}^{S}=i\omega\rho\iint_{S_{B}}\phi^{S}n_{\mu} \,dS

</math></center>

where the time-dependence has been removed in both cases. By expressing the radiation potential as a sum over the individual modes, the radiation force can written as <math>\sum_{\nu} v_{\nu}f_{\nu\mu}</math> where

<center><math>

f_{\nu\mu}=i\omega\rho\iint_{\Gamma} \phi_{\nu}n_{\mu} \,dS.

</math></center>

It is conventional to decompose the radiation force into a term featuring the added mass matrix <math>a_{\alpha\beta}</math> and a term featuring the damping matrix <math>b_{\alpha\beta}</math> as follows

<center><math>(7)

f_{\beta\alpha}=i\omega(a_{\beta\alpha}+\frac{ib_{\beta\alpha}}{\omega})

</math></center>

where the added mass and damping coefficients are in phase with the acceleration and velocity respectively. These real quantities depend only on the geometry of the structure and the oscillation frequency <math>\omega</math> and describe many of the properties of the structure. In particular, (as will be shown see later) if the variation of these coefficients with frequency is known then we can predict the resonant behaviour of the structure in the time-domain as well as the frequency domain. The computation of these quantities was considered to be very important in marine engineering and naval architecture and so a variety of different computer codes were developed to compute the coefficients efficiently. The boundary element method forms the basis of most of the computer code algorithms, the most widely-used example being WAMIT (see http:<math>\backslash\backslash</math>www.wamit.com) which determines frequency domain solutions for prescribed structures and structure motions.

==Frequency-domain equation of motion==

The radiation force coefficients also play a significant role in determining the motion of a floating structure possibly subject to mooring and applied forces. To obtain the correct equation of motion in the frequency domain it is important to adopt the approach similar to that taken by [[McIver 2005]], i.e. keep the initial condition terms in the Fourier transform operations. Therefore, if the Fourier transform of the velocity is <math>v_{\mu}(\omega)</math> then the Fourier transform of the acceleration is given by

<center><math>

\int_{0}^{\infty}\ddot{X}_{\mu}(t)e^{i\omega t}dt=-i\omega v_{\mu}(\omega) - V_{\mu}(0)

</math></center>

the time-derivative of the potential obeys a similar relation

<center><math>

\int_{0}^{\infty}\frac{\partial\Phi}{\partial t}e^{i\omega t}dt=-i\omega \phi_{\mu}(\omega) - \Phi(\mathbf{x},0).

</math></center>

Therefore, the ini\partial conditions of the structure <math>(X_{\mu}(0),V_{\mu}(0))</math> and of the potential <math>\Phi(\mathbf{x},0)</math> will be present in the Fourier transform of the equation motion which will be referred to as the frequency-domain equation of motion. If the frequency domain potential is decomposed in the same manner as before then the Fourier transform of the equation of motion for the structure is
<center><math>
M\left[-V_{\mu}(0)-i\omega v_{\mu}(\omega)\right]
=-\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS+F^{S}_{\mu}(\omega)+
i\omega \sum_{\nu} (f_{\mu\nu}(\omega)+i\gamma_{\mu\nu}/\omega)v_{\nu}(\omega)-
\sum_{\nu} c_{\mu\nu}\frac{v(\omega)+X(0)}{-i\omega}+f^{A}_{\mu}(\omega)
</math></center>
for <math>\mu=1,\ldots,6</math>, where <math>f^{A}_{\mu}(\omega)</math> is the Fourier transform of the applied force <math>F_{\mu}(t)</math> in equation~(\ref{linearisedmotion}). Although it is assumed that <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math>, for a non-zero initial velocity <math>\lim_{t^{+}\rightarrow 0}\Phi(\mathbf{x},t)\neq 0</math> because of the impulsive pressure on the free-surface resulting from the initial motion of the pressure. Therefore, the first term on the right-hand side of the above equation is not identically zero, rather is given by
<center><math>
\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS=a(\infty)V(0)
</math></center>
[[Mei 1983]]
where <math>a(\infty)=\lim_{\omega\rightarrow\infty}a(\omega)</math> is the infinite frequency added mass.

The frequency-domain equation is usually re-expressed in the following form
<center><math>
\sum_{\nu} \{c_{\mu\nu}-\omega^{2}\left[M_{\mu\mu}\delta_{\mu\nu}+a_{\mu\nu}(\omega)+i(b_{\mu\nu}+\gamma_{\mu\nu})/\omega\right]\}v_{\nu}(\omega)
= -i\omega\left[ F_{\mu}^{S}(\omega)+f_{\mu}(\omega)+(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)\right]-\sum_{\nu}c_{\mu\nu}X_{\nu}(0)
</math></center>
for <math>\mu=1,\ldots,6</math>. If it is assumed that the structure can only move in one mode of motion and that there is no incident wave or applied force then the complex velocity will be given by
<center><math>
v_{\mu}(\omega)=\frac{-i\omega(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)-c_{\mu\mu}X_{\mu}(0)}{c_{\mu\mu}-\omega^{2}\left[M_{\mu\mu}+a_{\mu\mu}(\omega)+i(b_{\mu\mu}+\gamma_{\mu\mu})/\omega\right]}.
</math></center>

As described by [[McIver 2006]], the zero of the denominator gives the location of the pole referred to as the motion resonance. The complex force coefficient <math>f_{\mu\nu}(\omega)</math> will also possess a pole, however this resonance is annulled by the frequency-dependence of the complex velocity (see [[McIver 2006]]) and so the motion, in the absence of incident waves, is dominated by the motion resonance term. Solving for <math>v(\omega)</math> then the time-domain velocity can be recovered using the inverse transform
<center><math>
V(t)=\frac{1}{\pi} Re \int^{\infty}_{0}v(\omega)e^{-i\omega t} dt.
</math></center>

==Integro-differential equation method==

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by <math>\delta(t-\tau)V_{\alpha}(\tau)</math> where <math>\dot{X_{\alpha}}=V_{\alpha}</math>. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function <math>V_{\alpha}(t)</math> an integral of the impulse response over the range <math>-\infty<\tau<\infty</math> must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

<center><math>(11)

\mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau

</math></center>

where <math>m_{\beta\alpha}(\infty)</math> is an infinite frequency added mass coefficient and <math>L_{\beta\alpha}(t-\tau)</math> is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. <math>V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \}</math>, the impulse response function <math>L_{\beta\alpha}</math> can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

<center><math>

\begin{split}

a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, d\tau \

b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, d\tau

\end{split}

</math></center>

and so if the added mass or damping coefficients are known for all frequencies <math>0\leq\omega\leq\infty</math>, the impulse response or memory function <math>L_{\alpha\beta}(t)</math> can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

<center><math>

\left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t),

</math></center>
for <math>\alpha=1,\ldots,6</math>, where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency <math>\omega</math> on the interval <math>[0,\infty)</math>. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

==Integro-differential equation method==

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by <math>\delta(t-\tau)V_{\alpha}(\tau)</math> where <math>\dot{X_{\alpha}}=V_{\alpha}</math>. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function <math>V_{\alpha}(t)</math> an integral of the impulse response over the range <math>-\infty<\tau<\infty</math> must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

<center><math>(11)

\mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau

</math></center>

where <math>m_{\beta\alpha}(\infty)</math> is an infinite frequency added mass coefficient and <math>L_{\beta\alpha}(t-\tau)</math> is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. <math>V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \}</math>, the impulse response function <math>L_{\beta\alpha}</math> can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

<center><math>

\begin{split}

a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, d\tau \

b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, d\tau

\end{split}

</math></center>

and so if the added mass or damping coefficients are known for all frequencies <math>0\leq\omega\leq\infty</math>, the impulse response or memory function <math>L_{\alpha\beta}(t)</math> can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

<center><math>

\left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t),

</math></center>
for <math>\alpha=1,\ldots,6</math>, where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency <math>\omega</math> on the interval <math>[0,\infty)</math>. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

Template:Initial conditions floating body

2009-09-03T01:56:23Z

Cfitzgerald:

The initial generalised displacements <math>\xi_{\mu}</math> and velocities <math>\dot{\xi}_{\mu}</math> of the body must be specified for all modes <math>\mu</math>

Category:Time-Dependent Linear Water Waves

2009-09-03T01:55:11Z

Cfitzgerald: /* Floating body constrained to move in heave */

[[Category:Linear Water-Wave Theory]]

{{incomplete pages}}

Generally the focus of research is on the [[Frequency Domain Problem]]. The time-domain
problem can be solved by [[Generalized Eigenfunction Expansion]] or by
an [[Expansion over the Resonances]] or using [[Memory Effect Function]].

== Introduction ==

Analytical solutions of the linearised time-domain equations for a coupled-motion or forced-motion problems are very rare. In fact, only a small number of analytic time-domain solutions have been obtained (see the papers by [[Kennard 1949]] and [[McIver 1994]]) for structures with simple geometries. Therefore, a variety of numerical methods have been developed to solve initial-value time-domain wave-structure interaction problems. The Fourier transform enables the time and frequency domain quantities to be related and the frequency domain forces and potentials have important roles in the development of the numerical time-domain solution methods. This can be partly attributed to the fact that, in the past, analytical or semi-analytical solutions to linearised water-wave problems were more easily obtained in the frequency domain given that the assumption of time-harmonic motion results in a significant simplification of the interaction equations. Therefore, it was natural to develop time-domain solutions based on frequency domain results and the Fourier transform relation between the domains.

== Equations of motion in the Time Domain ==

=== Two Dimensional Equations for fixed bodies in the time domain ===

{{Equations for fixed bodies in the time domain}}

=== Two dimensional equations for a floating body ===
We now consider the equations for a floating two-dimensional structure.
{{equations of motion time domain without body condition}}

The body boundary condition for a fixed body is given in terms
of the 3 rigid body motions, namely surge, heave and pitch which are indexed as <math>\mu=1,3,5</math> in order to be consistent with the three-dimensional problem. We have a kinematic condition
<center><math>
\partial_{n}\Phi=\sum\partial_t \xi_{\nu}\mathbf{n}_{\nu},\ \mathbf{x}\in\partial\Omega,
</math></center>
where <math>\xi_{\nu}</math> is the motion of the <math>\mu</math>th mode and
<math>\mathbf{n}_{\nu}</math> is the normal associated with this mode.
The dynamic condition is the equation of motion for the structure:
<center><math>
M_{\mu\nu}\partial_t^2 \xi_{\nu}=-\rho\iint_{\partial\Omega}\partial_t\Phi n_{\mu}\, dS- c_{\mu\nu}\xi_{\nu},\quad \textrm{for} \qquad \mu=1,3,5,
</math></center>
where summation over repeated indices is assumed. In this equation, <math>M_{\mu\nu}</math> are the elements of the mass matrix
<center><math>
\mathbf{M}=\left[
\begin{matrix}
M & 0 & M(z^c-Z^R) \\
0 & M & -M(x^c-X^R) \\
M(z^c-Z^R)& -M(x^c-X^R) & I^b_{11}+I^b_{33}
\end{matrix}
\right] ,
</math></center>
for the structure and <math>c_{\mu\nu}</math> are the elements of the buoyancy matrix
<center><math>
\mathbf{C}=\left[
\begin{matrix}
0 & 0 & 0 \\
0 & \rho g W & -\rho g I^A_{1} \\
0 & -\rho g I^A_{1} & \begin{matrix}\rho g (I^A_{11}+I^V_3)-\\
Mg(z^c-Z^R) \end{matrix}
\end{matrix}
\right].
</math></center>
The terms <math>I^b_{11}</math>, <math>I^b_{33}</math> are the moments of inertia of the body about the <math>x</math> and <math>z</math> axes and the terms <math>I_1^{A}</math>, <math>I^{A}_{11}</math> are the first and second moments of the waterplane (the waterplane area is denoted <math>W</math>) about the <math>x</math>-axis (see Chapter 7, [[Mei 1983]]). In addition, <math>(x^c,z^c)</math> and <math>(X^R,Z^R)</math> are the positions of the centre of mass and centre of rotation of the body and <math>I^{V}_{3}</math> is <math>z</math>-component centre of buoyancy of the structure. Thus, the coupled equations of motion for a floating structure have been derived. (N.B. if is assumed that the centre of rotation and the centre of mass of the structure coincide, i.e. if it is assumed that the body is semi-submerged, the mass and buoyancy matrices become diagonal). Any wave incidence is assumed to be propagating in the positive <math>x</math> direction.) The scattering and radiation problems are simpler than the coupled problem because the motion of the the structure is then prescribed.

The initial conditions are
{{initial free surface time domain}}
and
{{initial conditions floating body}}

=== Floating body constrained to move in heave ===
The simplest type of floating body problem concerns the motion of a body constrained to move in heave in two-dimensions. Apart from the boundary condition on the structure surface which becomes
<center><math>
\partial_n \Phi = \partial_t\xi_{3}\mathbf{n}_{3},
</math></center>
where <math>\xi_{3}</math> is the heave displacement of the structure, the equations governing the motion of the fluid remain the same. The equation of motion for a body constrained to move in heave only is
<center><math>
M\partial_t^2 \xi_{3}=-\rho\iint_{\partial\Omega}\partial_t\Phi n_{3}\, dS- \rho g W\xi_{3}.
</math></center>
which is a significant simplification compared to the general case - the mass matrix and buoyancy matrix are replaced simply by the mass <math>M</math> and the hydrostatic term <math>\rho g W</math>. The initial conditions for the fluid and the structure (<math>\xi_{3}(0)</math>, <math>\partial_t\xi_{3}(0)</math>) must also be prescribed to complete the problem specification.

==Frequency domain definitions==

Linearised water-wave problems were first investigated in the frequency domain wherein all motions are assumed to be time-harmonic and to have begun from <math>t= -\infty</math> . The time-dependence can be removed in a straightforward manner by substituting for the potential and any structural motions using
<center><math>
\Phi(\mathbf{x},t)=Re \{\phi(\mathbf{x},\omega) e^{-i\omega t}\}
</math></center>
<center><math>
V(\mathbf{x},t)=Re \{ v(\mathbf{x},\omega) e^{-i\omega t}\}
</math></center>
and this results in a considerable simplification of the problem. The complex amplitudes of the dynamic quantities vary with frequency and by removing the time-dependence the governing equations and boundary conditions will feature these terms only.
However, \citeasnoun{mciver2003} have shown that the frequency-domain potential can also be defined using the following operations on the time-domain solution:

*take a Laplace transform of the time-domain potential
<center><math>
\hat{\phi}(\mathbf{x},s)=\int^{\infty}_{0}\Phi(\mathbf{x},t)e^{-s t}\, dt ,\quad \textrm{Re } s >0;
</math></center>

*apply the change of variables <math>s=-i\omega</math> so that
<center><math>
\phi(\mathbf{x},\omega)=\hat{\phi}(\mathbf{x},-i\omega).
</math></center>

*noting that <math>\phi(\mathbf{x},-\omega)=\bar{\phi}(\mathbf{x},\omega)</math>, the inverse Fourier transform is given by
<center><math>(3)
\Phi(\mathbf{x},t)=\frac{1}{2\pi}\int^{\infty}_{-\infty}\phi(\mathbf{x},\omega)e^{-i\omega t} d\omega =\frac{1}{\pi}Re\int^{\infty}_{0}\phi(\mathbf{x},\omega)e^{-i\omega t}\,d\omega
</math></center>
where the path of integration must pass over any singularities of <math>\phi</math> that lie on the real axis and it has been assumed that there is no motion prior to <math>t=0</math>, i.e. <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math>.

If <math>\omega</math> is real then <math>\phi(\mathbf{x},\omega)</math> is just the frequency domain potential. By the principle of causality, which precludes the existence of the response before the cause so that <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math> , no singularities can exist in <math>\textrm{Im } \omega>0</math>. Therefore, if it is assumed that no singularities exist on the real axis the inverse transform will integrate the frequency-domain potential over the full range of <math>\omega</math>. With this approach, the frequency-domain equations are obtained by applying the Fourier transform to the time-dependent equations. However, unlike the equations resulting from the assumptions~(2) of time-harmonic motion from <math>t= -\infty</math>, the initial conditions will be present in the frequency-domain equations arising from the Fourier transformation. Although, the latter equations are more mathematically consistent it is usual to set all initial conditions to zero before attempting to solve the equations even if this is not the case. Nevertheless, in the frequency-domain analysis of resonances in coupled motion problems \citeasnoun{pmciver2005} retains the initial condition terms in the equation of motion for the structure and obtains some important results regarding the resonant behaviour of the motion. Therefore, the retention of the initial condition terms depends on the application of the subsequent equations.

Many analytical and approximate solutions exist for frequency-domain problems involving structures with simple geometries. Thus, in the simplest frequency-domain radiation and diffraction problems it is possible to completely determine the velocity potential analytically. The hydrodynamic forces, which are crucial in the context of marine engineering, can then be calculated directly from the potential. For more general geometries this is not possible and the hydrodynamic forces must be computed using numerical methods. Nevertheless, it is useful to consider the analytical expressions for the various potentials and forces in the frequency-domain problem.

In a linearised coupled motion problem, it is customary to decompose the total velocity potential into a scattering potential <math>\phi^{S}</math> and a radiation potential <math>\phi^{R}</math>. Thus, the coupled motion problem will require the solution of the radiation problem and the scattering problem. The radiation potential can be further decomposed into a sum over the six modes of motion so that the total frequency domain potential is
<center><math>
\phi=\phi^{S}+\sum_{\mu}v_{\mu}\phi_{\mu}
</math></center>

where <math>u_{\mu}</math> is the complex amplitude of the generalised velocity in the <math>\mu</math> direction and <math>\phi_{\mu}</math> describes the fluid response due to the forced oscillations in mode <math>\mu</math> with unit velocity amplitude. The velocities in each mode will be determined from the frequency domain equation of motion for the body. However, this requires the computation of the radiation and exciting forces on the structure. Therefore, it is necessary to consider the radiation problem prior to the coupled motion equation. It is first important to note that, for a given mode <math>\mu</math> the boundary condition on the structure will be

<center><math>

\frac{\partial\phi_{\mu}}{\partial n}=n_{\mu},

</math></center>

because the total velocity is

<center><math>

v(\omega)=\sum_{\mu}v_{\mu}n_{\mu}

</math></center>

where <math>n_{\mu}</math> is the <math>\mu</math> component of the generalised normal and <math>v_{\mu}</math> is the component of the generalised complex velocity amplitude in this direction. The hydrodynamic force on the structure in the <math>\mu</math> direction due to the fluid response to the forced oscillations is

<center><math>(5)

F_{\mu}^{R}=i\omega\rho\iint_{S_{B}}\phi^{R}n_{\mu} \,dS

</math></center>

and the force due to the diffraction of an incident wave by the fixed structure is

<center><math>(6)

F_{\mu}^{S}=i\omega\rho\iint_{S_{B}}\phi^{S}n_{\mu} \,dS

</math></center>

where the time-dependence has been removed in both cases. By expressing the radiation potential as a sum over the individual modes, the radiation force can written as <math>\sum_{\nu} v_{\nu}f_{\nu\mu}</math> where

<center><math>

f_{\nu\mu}=i\omega\rho\iint_{\Gamma} \phi_{\nu}n_{\mu} \,dS.

</math></center>

It is conventional to decompose the radiation force into a term featuring the added mass matrix <math>a_{\alpha\beta}</math> and a term featuring the damping matrix <math>b_{\alpha\beta}</math> as follows

<center><math>(7)

f_{\beta\alpha}=i\omega(a_{\beta\alpha}+\frac{ib_{\beta\alpha}}{\omega})

</math></center>

where the added mass and damping coefficients are in phase with the acceleration and velocity respectively. These real quantities depend only on the geometry of the structure and the oscillation frequency <math>\omega</math> and describe many of the properties of the structure. In particular, (as will be shown see later) if the variation of these coefficients with frequency is known then we can predict the resonant behaviour of the structure in the time-domain as well as the frequency domain. The computation of these quantities was considered to be very important in marine engineering and naval architecture and so a variety of different computer codes were developed to compute the coefficients efficiently. The boundary element method forms the basis of most of the computer code algorithms, the most widely-used example being WAMIT (see http:<math>\backslash\backslash</math>www.wamit.com) which determines frequency domain solutions for prescribed structures and structure motions.

==Frequency-domain equation of motion==

The radiation force coefficients also play a significant role in determining the motion of a floating structure possibly subject to mooring and applied forces. To obtain the correct equation of motion in the frequency domain it is important to adopt the approach similar to that taken by [[McIver 2005]], i.e. keep the initial condition terms in the Fourier transform operations. Therefore, if the Fourier transform of the velocity is <math>v_{\mu}(\omega)</math> then the Fourier transform of the acceleration is given by

<center><math>

\int_{0}^{\infty}\ddot{X}_{\mu}(t)e^{i\omega t}dt=-i\omega v_{\mu}(\omega) - V_{\mu}(0)

</math></center>

the time-derivative of the potential obeys a similar relation

<center><math>

\int_{0}^{\infty}\frac{\partial\Phi}{\partial t}e^{i\omega t}dt=-i\omega \phi_{\mu}(\omega) - \Phi(\mathbf{x},0).

</math></center>

Therefore, the ini\partial conditions of the structure <math>(X_{\mu}(0),V_{\mu}(0))</math> and of the potential <math>\Phi(\mathbf{x},0)</math> will be present in the Fourier transform of the equation motion which will be referred to as the frequency-domain equation of motion. If the frequency domain potential is decomposed in the same manner as before then the Fourier transform of the equation of motion for the structure is
<center><math>
M\left[-V_{\mu}(0)-i\omega v_{\mu}(\omega)\right]
=-\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS+F^{S}_{\mu}(\omega)+
i\omega \sum_{\nu} (f_{\mu\nu}(\omega)+i\gamma_{\mu\nu}/\omega)v_{\nu}(\omega)-
\sum_{\nu} c_{\mu\nu}\frac{v(\omega)+X(0)}{-i\omega}+f^{A}_{\mu}(\omega)
</math></center>
for <math>\mu=1,\ldots,6</math>, where <math>f^{A}_{\mu}(\omega)</math> is the Fourier transform of the applied force <math>F_{\mu}(t)</math> in equation~(\ref{linearisedmotion}). Although it is assumed that <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math>, for a non-zero initial velocity <math>\lim_{t^{+}\rightarrow 0}\Phi(\mathbf{x},t)\neq 0</math> because of the impulsive pressure on the free-surface resulting from the initial motion of the pressure. Therefore, the first term on the right-hand side of the above equation is not identically zero, rather is given by
<center><math>
\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS=a(\infty)V(0)
</math></center>
[[Mei 1983]]
where <math>a(\infty)=\lim_{\omega\rightarrow\infty}a(\omega)</math> is the infinite frequency added mass.

The frequency-domain equation is usually re-expressed in the following form
<center><math>
\sum_{\nu} \{c_{\mu\nu}-\omega^{2}\left[M_{\mu\mu}\delta_{\mu\nu}+a_{\mu\nu}(\omega)+i(b_{\mu\nu}+\gamma_{\mu\nu})/\omega\right]\}v_{\nu}(\omega)
= -i\omega\left[ F_{\mu}^{S}(\omega)+f_{\mu}(\omega)+(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)\right]-\sum_{\nu}c_{\mu\nu}X_{\nu}(0)
</math></center>
for <math>\mu=1,\ldots,6</math>. If it is assumed that the structure can only move in one mode of motion and that there is no incident wave or applied force then the complex velocity will be given by
<center><math>
v_{\mu}(\omega)=\frac{-i\omega(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)-c_{\mu\mu}X_{\mu}(0)}{c_{\mu\mu}-\omega^{2}\left[M_{\mu\mu}+a_{\mu\mu}(\omega)+i(b_{\mu\mu}+\gamma_{\mu\mu})/\omega\right]}.
</math></center>

As described by [[McIver 2006]], the zero of the denominator gives the location of the pole referred to as the motion resonance. The complex force coefficient <math>f_{\mu\nu}(\omega)</math> will also possess a pole, however this resonance is annulled by the frequency-dependence of the complex velocity (see [[McIver 2006]]) and so the motion, in the absence of incident waves, is dominated by the motion resonance term. Solving for <math>v(\omega)</math> then the time-domain velocity can be recovered using the inverse transform
<center><math>
V(t)=\frac{1}{\pi} Re \int^{\infty}_{0}v(\omega)e^{-i\omega t} dt.
</math></center>

==Integro-differential equation method==

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by <math>\delta(t-\tau)V_{\alpha}(\tau)</math> where <math>\dot{X_{\alpha}}=V_{\alpha}</math>. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function <math>V_{\alpha}(t)</math> an integral of the impulse response over the range <math>-\infty<\tau<\infty</math> must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

<center><math>(11)

\mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau

</math></center>

where <math>m_{\beta\alpha}(\infty)</math> is an infinite frequency added mass coefficient and <math>L_{\beta\alpha}(t-\tau)</math> is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. <math>V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \}</math>, the impulse response function <math>L_{\beta\alpha}</math> can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

<center><math>

\begin{split}

a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, d\tau \

b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, d\tau

\end{split}

</math></center>

and so if the added mass or damping coefficients are known for all frequencies <math>0\leq\omega\leq\infty</math>, the impulse response or memory function <math>L_{\alpha\beta}(t)</math> can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

<center><math>

\left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t),

</math></center>
for <math>\alpha=1,\ldots,6</math>, where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency <math>\omega</math> on the interval <math>[0,\infty)</math>. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

==Integro-differential equation method==

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by <math>\delta(t-\tau)V_{\alpha}(\tau)</math> where <math>\dot{X_{\alpha}}=V_{\alpha}</math>. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function <math>V_{\alpha}(t)</math> an integral of the impulse response over the range <math>-\infty<\tau<\infty</math> must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

<center><math>(11)

\mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau

</math></center>

where <math>m_{\beta\alpha}(\infty)</math> is an infinite frequency added mass coefficient and <math>L_{\beta\alpha}(t-\tau)</math> is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. <math>V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \}</math>, the impulse response function <math>L_{\beta\alpha}</math> can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

<center><math>

\begin{split}

a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, d\tau \

b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, d\tau

\end{split}

</math></center>

and so if the added mass or damping coefficients are known for all frequencies <math>0\leq\omega\leq\infty</math>, the impulse response or memory function <math>L_{\alpha\beta}(t)</math> can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

<center><math>

\left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t),

</math></center>
for <math>\alpha=1,\ldots,6</math>, where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency <math>\omega</math> on the interval <math>[0,\infty)</math>. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

Category:Time-Dependent Linear Water Waves

2009-09-03T01:53:00Z

Cfitzgerald: /* Floating body constrained to move in heave */

[[Category:Linear Water-Wave Theory]]

{{incomplete pages}}

Generally the focus of research is on the [[Frequency Domain Problem]]. The time-domain
problem can be solved by [[Generalized Eigenfunction Expansion]] or by
an [[Expansion over the Resonances]] or using [[Memory Effect Function]].

== Introduction ==

Analytical solutions of the linearised time-domain equations for a coupled-motion or forced-motion problems are very rare. In fact, only a small number of analytic time-domain solutions have been obtained (see the papers by [[Kennard 1949]] and [[McIver 1994]]) for structures with simple geometries. Therefore, a variety of numerical methods have been developed to solve initial-value time-domain wave-structure interaction problems. The Fourier transform enables the time and frequency domain quantities to be related and the frequency domain forces and potentials have important roles in the development of the numerical time-domain solution methods. This can be partly attributed to the fact that, in the past, analytical or semi-analytical solutions to linearised water-wave problems were more easily obtained in the frequency domain given that the assumption of time-harmonic motion results in a significant simplification of the interaction equations. Therefore, it was natural to develop time-domain solutions based on frequency domain results and the Fourier transform relation between the domains.

== Equations of motion in the Time Domain ==

=== Two Dimensional Equations for fixed bodies in the time domain ===

{{Equations for fixed bodies in the time domain}}

=== Two dimensional equations for a floating body ===
We now consider the equations for a floating two-dimensional structure.
{{equations of motion time domain without body condition}}

The body boundary condition for a fixed body is given in terms
of the 3 rigid body motions, namely surge, heave and pitch which are indexed as <math>\mu=1,3,5</math> in order to be consistent with the three-dimensional problem. We have a kinematic condition
<center><math>
\partial_{n}\Phi=\sum\partial_t \xi_{\nu}\mathbf{n}_{\nu},\ \mathbf{x}\in\partial\Omega,
</math></center>
where <math>\xi_{\nu}</math> is the motion of the <math>\mu</math>th mode and
<math>\mathbf{n}_{\nu}</math> is the normal associated with this mode.
The dynamic condition is the equation of motion for the structure:
<center><math>
M_{\mu\nu}\partial_t^2 \xi_{\nu}=-\rho\iint_{\partial\Omega}\partial_t\Phi n_{\mu}\, dS- c_{\mu\nu}\xi_{\nu},\quad \textrm{for} \qquad \mu=1,3,5,
</math></center>
where summation over repeated indices is assumed. In this equation, <math>M_{\mu\nu}</math> are the elements of the mass matrix
<center><math>
\mathbf{M}=\left[
\begin{matrix}
M & 0 & M(z^c-Z^R) \\
0 & M & -M(x^c-X^R) \\
M(z^c-Z^R)& -M(x^c-X^R) & I^b_{11}+I^b_{33}
\end{matrix}
\right] ,
</math></center>
for the structure and <math>c_{\mu\nu}</math> are the elements of the buoyancy matrix
<center><math>
\mathbf{C}=\left[
\begin{matrix}
0 & 0 & 0 \\
0 & \rho g W & -\rho g I^A_{1} \\
0 & -\rho g I^A_{1} & \begin{matrix}\rho g (I^A_{11}+I^V_3)-\\
Mg(z^c-Z^R) \end{matrix}
\end{matrix}
\right].
</math></center>
The terms <math>I^b_{11}</math>, <math>I^b_{33}</math> are the moments of inertia of the body about the <math>x</math> and <math>z</math> axes and the terms <math>I_1^{A}</math>, <math>I^{A}_{11}</math> are the first and second moments of the waterplane (the waterplane area is denoted <math>W</math>) about the <math>x</math>-axis (see Chapter 7, [[Mei 1983]]). In addition, <math>(x^c,z^c)</math> and <math>(X^R,Z^R)</math> are the positions of the centre of mass and centre of rotation of the body and <math>I^{V}_{3}</math> is <math>z</math>-component centre of buoyancy of the structure. Thus, the coupled equations of motion for a floating structure have been derived. (N.B. if is assumed that the centre of rotation and the centre of mass of the structure coincide, i.e. if it is assumed that the body is semi-submerged, the mass and buoyancy matrices become diagonal). Any wave incidence is assumed to be propagating in the positive <math>x</math> direction.) The scattering and radiation problems are simpler than the coupled problem because the motion of the the structure is then prescribed.

The initial conditions are
{{initial free surface time domain}}
and
{{initial conditions floating body}}

=== Floating body constrained to move in heave ===
The simplest type of floating body problem concerns the motion of a body constrained to move in heave in two-dimensions. Apart from the boundary condition on the structure surface which becomes
<center><math>
\partial_n \Phi = \partial_t\xi_{3}\mathbf{n}_{3},
</math></center>
where <math>\xi_{3}</math> is the heave displacement of the structure, the equations governing the motion of the fluid remain the same. The equation of motion for a body constrained to move in heave only is
<center><math>
M\partial_t^2 \xi_{3}=-\rho\iint_{\partial\Omega}\partial_t\Phi n_{3}\, dS- \rho g W\xi_{3}.
</math></center>
which is a significant simplification compared to the general case - the mass matrix and buoyancy matrix are replaced simply by the mass <math>M</math> and the hydrostatic term <math>\rho g W</math>.

==Frequency domain definitions==

Linearised water-wave problems were first investigated in the frequency domain wherein all motions are assumed to be time-harmonic and to have begun from <math>t= -\infty</math> . The time-dependence can be removed in a straightforward manner by substituting for the potential and any structural motions using
<center><math>
\Phi(\mathbf{x},t)=Re \{\phi(\mathbf{x},\omega) e^{-i\omega t}\}
</math></center>
<center><math>
V(\mathbf{x},t)=Re \{ v(\mathbf{x},\omega) e^{-i\omega t}\}
</math></center>
and this results in a considerable simplification of the problem. The complex amplitudes of the dynamic quantities vary with frequency and by removing the time-dependence the governing equations and boundary conditions will feature these terms only.
However, \citeasnoun{mciver2003} have shown that the frequency-domain potential can also be defined using the following operations on the time-domain solution:

*take a Laplace transform of the time-domain potential
<center><math>
\hat{\phi}(\mathbf{x},s)=\int^{\infty}_{0}\Phi(\mathbf{x},t)e^{-s t}\, dt ,\quad \textrm{Re } s >0;
</math></center>

*apply the change of variables <math>s=-i\omega</math> so that
<center><math>
\phi(\mathbf{x},\omega)=\hat{\phi}(\mathbf{x},-i\omega).
</math></center>

*noting that <math>\phi(\mathbf{x},-\omega)=\bar{\phi}(\mathbf{x},\omega)</math>, the inverse Fourier transform is given by
<center><math>(3)
\Phi(\mathbf{x},t)=\frac{1}{2\pi}\int^{\infty}_{-\infty}\phi(\mathbf{x},\omega)e^{-i\omega t} d\omega =\frac{1}{\pi}Re\int^{\infty}_{0}\phi(\mathbf{x},\omega)e^{-i\omega t}\,d\omega
</math></center>
where the path of integration must pass over any singularities of <math>\phi</math> that lie on the real axis and it has been assumed that there is no motion prior to <math>t=0</math>, i.e. <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math>.

If <math>\omega</math> is real then <math>\phi(\mathbf{x},\omega)</math> is just the frequency domain potential. By the principle of causality, which precludes the existence of the response before the cause so that <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math> , no singularities can exist in <math>\textrm{Im } \omega>0</math>. Therefore, if it is assumed that no singularities exist on the real axis the inverse transform will integrate the frequency-domain potential over the full range of <math>\omega</math>. With this approach, the frequency-domain equations are obtained by applying the Fourier transform to the time-dependent equations. However, unlike the equations resulting from the assumptions~(2) of time-harmonic motion from <math>t= -\infty</math>, the initial conditions will be present in the frequency-domain equations arising from the Fourier transformation. Although, the latter equations are more mathematically consistent it is usual to set all initial conditions to zero before attempting to solve the equations even if this is not the case. Nevertheless, in the frequency-domain analysis of resonances in coupled motion problems \citeasnoun{pmciver2005} retains the initial condition terms in the equation of motion for the structure and obtains some important results regarding the resonant behaviour of the motion. Therefore, the retention of the initial condition terms depends on the application of the subsequent equations.

Many analytical and approximate solutions exist for frequency-domain problems involving structures with simple geometries. Thus, in the simplest frequency-domain radiation and diffraction problems it is possible to completely determine the velocity potential analytically. The hydrodynamic forces, which are crucial in the context of marine engineering, can then be calculated directly from the potential. For more general geometries this is not possible and the hydrodynamic forces must be computed using numerical methods. Nevertheless, it is useful to consider the analytical expressions for the various potentials and forces in the frequency-domain problem.

In a linearised coupled motion problem, it is customary to decompose the total velocity potential into a scattering potential <math>\phi^{S}</math> and a radiation potential <math>\phi^{R}</math>. Thus, the coupled motion problem will require the solution of the radiation problem and the scattering problem. The radiation potential can be further decomposed into a sum over the six modes of motion so that the total frequency domain potential is
<center><math>
\phi=\phi^{S}+\sum_{\mu}v_{\mu}\phi_{\mu}
</math></center>

where <math>u_{\mu}</math> is the complex amplitude of the generalised velocity in the <math>\mu</math> direction and <math>\phi_{\mu}</math> describes the fluid response due to the forced oscillations in mode <math>\mu</math> with unit velocity amplitude. The velocities in each mode will be determined from the frequency domain equation of motion for the body. However, this requires the computation of the radiation and exciting forces on the structure. Therefore, it is necessary to consider the radiation problem prior to the coupled motion equation. It is first important to note that, for a given mode <math>\mu</math> the boundary condition on the structure will be

<center><math>

\frac{\partial\phi_{\mu}}{\partial n}=n_{\mu},

</math></center>

because the total velocity is

<center><math>

v(\omega)=\sum_{\mu}v_{\mu}n_{\mu}

</math></center>

where <math>n_{\mu}</math> is the <math>\mu</math> component of the generalised normal and <math>v_{\mu}</math> is the component of the generalised complex velocity amplitude in this direction. The hydrodynamic force on the structure in the <math>\mu</math> direction due to the fluid response to the forced oscillations is

<center><math>(5)

F_{\mu}^{R}=i\omega\rho\iint_{S_{B}}\phi^{R}n_{\mu} \,dS

</math></center>

and the force due to the diffraction of an incident wave by the fixed structure is

<center><math>(6)

F_{\mu}^{S}=i\omega\rho\iint_{S_{B}}\phi^{S}n_{\mu} \,dS

</math></center>

where the time-dependence has been removed in both cases. By expressing the radiation potential as a sum over the individual modes, the radiation force can written as <math>\sum_{\nu} v_{\nu}f_{\nu\mu}</math> where

<center><math>

f_{\nu\mu}=i\omega\rho\iint_{\Gamma} \phi_{\nu}n_{\mu} \,dS.

</math></center>

It is conventional to decompose the radiation force into a term featuring the added mass matrix <math>a_{\alpha\beta}</math> and a term featuring the damping matrix <math>b_{\alpha\beta}</math> as follows

<center><math>(7)

f_{\beta\alpha}=i\omega(a_{\beta\alpha}+\frac{ib_{\beta\alpha}}{\omega})

</math></center>

where the added mass and damping coefficients are in phase with the acceleration and velocity respectively. These real quantities depend only on the geometry of the structure and the oscillation frequency <math>\omega</math> and describe many of the properties of the structure. In particular, (as will be shown see later) if the variation of these coefficients with frequency is known then we can predict the resonant behaviour of the structure in the time-domain as well as the frequency domain. The computation of these quantities was considered to be very important in marine engineering and naval architecture and so a variety of different computer codes were developed to compute the coefficients efficiently. The boundary element method forms the basis of most of the computer code algorithms, the most widely-used example being WAMIT (see http:<math>\backslash\backslash</math>www.wamit.com) which determines frequency domain solutions for prescribed structures and structure motions.

==Frequency-domain equation of motion==

The radiation force coefficients also play a significant role in determining the motion of a floating structure possibly subject to mooring and applied forces. To obtain the correct equation of motion in the frequency domain it is important to adopt the approach similar to that taken by [[McIver 2005]], i.e. keep the initial condition terms in the Fourier transform operations. Therefore, if the Fourier transform of the velocity is <math>v_{\mu}(\omega)</math> then the Fourier transform of the acceleration is given by

<center><math>

\int_{0}^{\infty}\ddot{X}_{\mu}(t)e^{i\omega t}dt=-i\omega v_{\mu}(\omega) - V_{\mu}(0)

</math></center>

the time-derivative of the potential obeys a similar relation

<center><math>

\int_{0}^{\infty}\frac{\partial\Phi}{\partial t}e^{i\omega t}dt=-i\omega \phi_{\mu}(\omega) - \Phi(\mathbf{x},0).

</math></center>

Therefore, the ini\partial conditions of the structure <math>(X_{\mu}(0),V_{\mu}(0))</math> and of the potential <math>\Phi(\mathbf{x},0)</math> will be present in the Fourier transform of the equation motion which will be referred to as the frequency-domain equation of motion. If the frequency domain potential is decomposed in the same manner as before then the Fourier transform of the equation of motion for the structure is
<center><math>
M\left[-V_{\mu}(0)-i\omega v_{\mu}(\omega)\right]
=-\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS+F^{S}_{\mu}(\omega)+
i\omega \sum_{\nu} (f_{\mu\nu}(\omega)+i\gamma_{\mu\nu}/\omega)v_{\nu}(\omega)-
\sum_{\nu} c_{\mu\nu}\frac{v(\omega)+X(0)}{-i\omega}+f^{A}_{\mu}(\omega)
</math></center>
for <math>\mu=1,\ldots,6</math>, where <math>f^{A}_{\mu}(\omega)</math> is the Fourier transform of the applied force <math>F_{\mu}(t)</math> in equation~(\ref{linearisedmotion}). Although it is assumed that <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math>, for a non-zero initial velocity <math>\lim_{t^{+}\rightarrow 0}\Phi(\mathbf{x},t)\neq 0</math> because of the impulsive pressure on the free-surface resulting from the initial motion of the pressure. Therefore, the first term on the right-hand side of the above equation is not identically zero, rather is given by
<center><math>
\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS=a(\infty)V(0)
</math></center>
[[Mei 1983]]
where <math>a(\infty)=\lim_{\omega\rightarrow\infty}a(\omega)</math> is the infinite frequency added mass.

The frequency-domain equation is usually re-expressed in the following form
<center><math>
\sum_{\nu} \{c_{\mu\nu}-\omega^{2}\left[M_{\mu\mu}\delta_{\mu\nu}+a_{\mu\nu}(\omega)+i(b_{\mu\nu}+\gamma_{\mu\nu})/\omega\right]\}v_{\nu}(\omega)
= -i\omega\left[ F_{\mu}^{S}(\omega)+f_{\mu}(\omega)+(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)\right]-\sum_{\nu}c_{\mu\nu}X_{\nu}(0)
</math></center>
for <math>\mu=1,\ldots,6</math>. If it is assumed that the structure can only move in one mode of motion and that there is no incident wave or applied force then the complex velocity will be given by
<center><math>
v_{\mu}(\omega)=\frac{-i\omega(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)-c_{\mu\mu}X_{\mu}(0)}{c_{\mu\mu}-\omega^{2}\left[M_{\mu\mu}+a_{\mu\mu}(\omega)+i(b_{\mu\mu}+\gamma_{\mu\mu})/\omega\right]}.
</math></center>

As described by [[McIver 2006]], the zero of the denominator gives the location of the pole referred to as the motion resonance. The complex force coefficient <math>f_{\mu\nu}(\omega)</math> will also possess a pole, however this resonance is annulled by the frequency-dependence of the complex velocity (see [[McIver 2006]]) and so the motion, in the absence of incident waves, is dominated by the motion resonance term. Solving for <math>v(\omega)</math> then the time-domain velocity can be recovered using the inverse transform
<center><math>
V(t)=\frac{1}{\pi} Re \int^{\infty}_{0}v(\omega)e^{-i\omega t} dt.
</math></center>

==Integro-differential equation method==

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by <math>\delta(t-\tau)V_{\alpha}(\tau)</math> where <math>\dot{X_{\alpha}}=V_{\alpha}</math>. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function <math>V_{\alpha}(t)</math> an integral of the impulse response over the range <math>-\infty<\tau<\infty</math> must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

<center><math>(11)

\mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau

</math></center>

where <math>m_{\beta\alpha}(\infty)</math> is an infinite frequency added mass coefficient and <math>L_{\beta\alpha}(t-\tau)</math> is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. <math>V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \}</math>, the impulse response function <math>L_{\beta\alpha}</math> can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

<center><math>

\begin{split}

a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, d\tau \

b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, d\tau

\end{split}

</math></center>

and so if the added mass or damping coefficients are known for all frequencies <math>0\leq\omega\leq\infty</math>, the impulse response or memory function <math>L_{\alpha\beta}(t)</math> can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

<center><math>

\left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t),

</math></center>
for <math>\alpha=1,\ldots,6</math>, where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency <math>\omega</math> on the interval <math>[0,\infty)</math>. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

==Integro-differential equation method==

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by <math>\delta(t-\tau)V_{\alpha}(\tau)</math> where <math>\dot{X_{\alpha}}=V_{\alpha}</math>. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function <math>V_{\alpha}(t)</math> an integral of the impulse response over the range <math>-\infty<\tau<\infty</math> must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

<center><math>(11)

\mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau

</math></center>

where <math>m_{\beta\alpha}(\infty)</math> is an infinite frequency added mass coefficient and <math>L_{\beta\alpha}(t-\tau)</math> is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. <math>V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \}</math>, the impulse response function <math>L_{\beta\alpha}</math> can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

<center><math>

\begin{split}

a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, d\tau \

b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, d\tau

\end{split}

</math></center>

and so if the added mass or damping coefficients are known for all frequencies <math>0\leq\omega\leq\infty</math>, the impulse response or memory function <math>L_{\alpha\beta}(t)</math> can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

<center><math>

\left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t),

</math></center>
for <math>\alpha=1,\ldots,6</math>, where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency <math>\omega</math> on the interval <math>[0,\infty)</math>. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

Category:Time-Dependent Linear Water Waves

2009-09-03T01:50:16Z

Cfitzgerald: /* Floating body constrained to move in heave */

[[Category:Linear Water-Wave Theory]]

{{incomplete pages}}

Generally the focus of research is on the [[Frequency Domain Problem]]. The time-domain
problem can be solved by [[Generalized Eigenfunction Expansion]] or by
an [[Expansion over the Resonances]] or using [[Memory Effect Function]].

== Introduction ==

Analytical solutions of the linearised time-domain equations for a coupled-motion or forced-motion problems are very rare. In fact, only a small number of analytic time-domain solutions have been obtained (see the papers by [[Kennard 1949]] and [[McIver 1994]]) for structures with simple geometries. Therefore, a variety of numerical methods have been developed to solve initial-value time-domain wave-structure interaction problems. The Fourier transform enables the time and frequency domain quantities to be related and the frequency domain forces and potentials have important roles in the development of the numerical time-domain solution methods. This can be partly attributed to the fact that, in the past, analytical or semi-analytical solutions to linearised water-wave problems were more easily obtained in the frequency domain given that the assumption of time-harmonic motion results in a significant simplification of the interaction equations. Therefore, it was natural to develop time-domain solutions based on frequency domain results and the Fourier transform relation between the domains.

== Equations of motion in the Time Domain ==

=== Two Dimensional Equations for fixed bodies in the time domain ===

{{Equations for fixed bodies in the time domain}}

=== Two dimensional equations for a floating body ===
We now consider the equations for a floating two-dimensional structure.
{{equations of motion time domain without body condition}}

The body boundary condition for a fixed body is given in terms
of the 3 rigid body motions, namely surge, heave and pitch which are indexed as <math>\mu=1,3,5</math> in order to be consistent with the three-dimensional problem. We have a kinematic condition
<center><math>
\partial_{n}\Phi=\sum\partial_t \xi_{\nu}\mathbf{n}_{\nu},\ \mathbf{x}\in\partial\Omega,
</math></center>
where <math>\xi_{\nu}</math> is the motion of the <math>\mu</math>th mode and
<math>\mathbf{n}_{\nu}</math> is the normal associated with this mode.
The dynamic condition is the equation of motion for the structure:
<center><math>
M_{\mu\nu}\partial_t^2 \xi_{\nu}=-\rho\iint_{\partial\Omega}\partial_t\Phi n_{\mu}\, dS- c_{\mu\nu}\xi_{\nu},\quad \textrm{for} \qquad \mu=1,3,5,
</math></center>
where summation over repeated indices is assumed. In this equation, <math>M_{\mu\nu}</math> are the elements of the mass matrix
<center><math>
\mathbf{M}=\left[
\begin{matrix}
M & 0 & M(z^c-Z^R) \\
0 & M & -M(x^c-X^R) \\
M(z^c-Z^R)& -M(x^c-X^R) & I^b_{11}+I^b_{33}
\end{matrix}
\right] ,
</math></center>
for the structure and <math>c_{\mu\nu}</math> are the elements of the buoyancy matrix
<center><math>
\mathbf{C}=\left[
\begin{matrix}
0 & 0 & 0 \\
0 & \rho g W & -\rho g I^A_{1} \\
0 & -\rho g I^A_{1} & \begin{matrix}\rho g (I^A_{11}+I^V_3)-\\
Mg(z^c-Z^R) \end{matrix}
\end{matrix}
\right].
</math></center>
The terms <math>I^b_{11}</math>, <math>I^b_{33}</math> are the moments of inertia of the body about the <math>x</math> and <math>z</math> axes and the terms <math>I_1^{A}</math>, <math>I^{A}_{11}</math> are the first and second moments of the waterplane (the waterplane area is denoted <math>W</math>) about the <math>x</math>-axis (see Chapter 7, [[Mei 1983]]). In addition, <math>(x^c,z^c)</math> and <math>(X^R,Z^R)</math> are the positions of the centre of mass and centre of rotation of the body and <math>I^{V}_{3}</math> is <math>z</math>-component centre of buoyancy of the structure. Thus, the coupled equations of motion for a floating structure have been derived. (N.B. if is assumed that the centre of rotation and the centre of mass of the structure coincide, i.e. if it is assumed that the body is semi-submerged, the mass and buoyancy matrices become diagonal). Any wave incidence is assumed to be propagating in the positive <math>x</math> direction.) The scattering and radiation problems are simpler than the coupled problem because the motion of the the structure is then prescribed.

The initial conditions are
{{initial free surface time domain}}
and
{{initial conditions floating body}}

=== Floating body constrained to move in heave ===
The simplest type of floating body problem concerns the motion of a body constrained to move in heave in two-dimensions. Apart from the boundary condition on the structure surface which becomes
<center><math>
\partial_n \Phi = \partial_t\xi_{3}\mathbf{n}_{3},
</math></center>
where <math>\xi_{3}</math> is the heave displacement of the structure, the equations governing the motion of the fluid remain the same. The equation of motion for a body constrained to move in heave only is
<center><math>
M\partial_t^2 \xi_{3}=-\rho\iint_{\partial\Omega}\partial_t\Phi n_{3}\, dS- \rho g W\xi_{3}.
</math></center>
which is a significant simplification compared to the general case. The mass matrix and buoyancy matrix are replaced by the mass <math>M</math> and the hydrostatic term <math>\rho g W</math>.

==Frequency domain definitions==

Linearised water-wave problems were first investigated in the frequency domain wherein all motions are assumed to be time-harmonic and to have begun from <math>t= -\infty</math> . The time-dependence can be removed in a straightforward manner by substituting for the potential and any structural motions using
<center><math>
\Phi(\mathbf{x},t)=Re \{\phi(\mathbf{x},\omega) e^{-i\omega t}\}
</math></center>
<center><math>
V(\mathbf{x},t)=Re \{ v(\mathbf{x},\omega) e^{-i\omega t}\}
</math></center>
and this results in a considerable simplification of the problem. The complex amplitudes of the dynamic quantities vary with frequency and by removing the time-dependence the governing equations and boundary conditions will feature these terms only.
However, \citeasnoun{mciver2003} have shown that the frequency-domain potential can also be defined using the following operations on the time-domain solution:

*take a Laplace transform of the time-domain potential
<center><math>
\hat{\phi}(\mathbf{x},s)=\int^{\infty}_{0}\Phi(\mathbf{x},t)e^{-s t}\, dt ,\quad \textrm{Re } s >0;
</math></center>

*apply the change of variables <math>s=-i\omega</math> so that
<center><math>
\phi(\mathbf{x},\omega)=\hat{\phi}(\mathbf{x},-i\omega).
</math></center>

*noting that <math>\phi(\mathbf{x},-\omega)=\bar{\phi}(\mathbf{x},\omega)</math>, the inverse Fourier transform is given by
<center><math>(3)
\Phi(\mathbf{x},t)=\frac{1}{2\pi}\int^{\infty}_{-\infty}\phi(\mathbf{x},\omega)e^{-i\omega t} d\omega =\frac{1}{\pi}Re\int^{\infty}_{0}\phi(\mathbf{x},\omega)e^{-i\omega t}\,d\omega
</math></center>
where the path of integration must pass over any singularities of <math>\phi</math> that lie on the real axis and it has been assumed that there is no motion prior to <math>t=0</math>, i.e. <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math>.

If <math>\omega</math> is real then <math>\phi(\mathbf{x},\omega)</math> is just the frequency domain potential. By the principle of causality, which precludes the existence of the response before the cause so that <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math> , no singularities can exist in <math>\textrm{Im } \omega>0</math>. Therefore, if it is assumed that no singularities exist on the real axis the inverse transform will integrate the frequency-domain potential over the full range of <math>\omega</math>. With this approach, the frequency-domain equations are obtained by applying the Fourier transform to the time-dependent equations. However, unlike the equations resulting from the assumptions~(2) of time-harmonic motion from <math>t= -\infty</math>, the initial conditions will be present in the frequency-domain equations arising from the Fourier transformation. Although, the latter equations are more mathematically consistent it is usual to set all initial conditions to zero before attempting to solve the equations even if this is not the case. Nevertheless, in the frequency-domain analysis of resonances in coupled motion problems \citeasnoun{pmciver2005} retains the initial condition terms in the equation of motion for the structure and obtains some important results regarding the resonant behaviour of the motion. Therefore, the retention of the initial condition terms depends on the application of the subsequent equations.

Many analytical and approximate solutions exist for frequency-domain problems involving structures with simple geometries. Thus, in the simplest frequency-domain radiation and diffraction problems it is possible to completely determine the velocity potential analytically. The hydrodynamic forces, which are crucial in the context of marine engineering, can then be calculated directly from the potential. For more general geometries this is not possible and the hydrodynamic forces must be computed using numerical methods. Nevertheless, it is useful to consider the analytical expressions for the various potentials and forces in the frequency-domain problem.

In a linearised coupled motion problem, it is customary to decompose the total velocity potential into a scattering potential <math>\phi^{S}</math> and a radiation potential <math>\phi^{R}</math>. Thus, the coupled motion problem will require the solution of the radiation problem and the scattering problem. The radiation potential can be further decomposed into a sum over the six modes of motion so that the total frequency domain potential is
<center><math>
\phi=\phi^{S}+\sum_{\mu}v_{\mu}\phi_{\mu}
</math></center>

where <math>u_{\mu}</math> is the complex amplitude of the generalised velocity in the <math>\mu</math> direction and <math>\phi_{\mu}</math> describes the fluid response due to the forced oscillations in mode <math>\mu</math> with unit velocity amplitude. The velocities in each mode will be determined from the frequency domain equation of motion for the body. However, this requires the computation of the radiation and exciting forces on the structure. Therefore, it is necessary to consider the radiation problem prior to the coupled motion equation. It is first important to note that, for a given mode <math>\mu</math> the boundary condition on the structure will be

<center><math>

\frac{\partial\phi_{\mu}}{\partial n}=n_{\mu},

</math></center>

because the total velocity is

<center><math>

v(\omega)=\sum_{\mu}v_{\mu}n_{\mu}

</math></center>

where <math>n_{\mu}</math> is the <math>\mu</math> component of the generalised normal and <math>v_{\mu}</math> is the component of the generalised complex velocity amplitude in this direction. The hydrodynamic force on the structure in the <math>\mu</math> direction due to the fluid response to the forced oscillations is

<center><math>(5)

F_{\mu}^{R}=i\omega\rho\iint_{S_{B}}\phi^{R}n_{\mu} \,dS

</math></center>

and the force due to the diffraction of an incident wave by the fixed structure is

<center><math>(6)

F_{\mu}^{S}=i\omega\rho\iint_{S_{B}}\phi^{S}n_{\mu} \,dS

</math></center>

where the time-dependence has been removed in both cases. By expressing the radiation potential as a sum over the individual modes, the radiation force can written as <math>\sum_{\nu} v_{\nu}f_{\nu\mu}</math> where

<center><math>

f_{\nu\mu}=i\omega\rho\iint_{\Gamma} \phi_{\nu}n_{\mu} \,dS.

</math></center>

It is conventional to decompose the radiation force into a term featuring the added mass matrix <math>a_{\alpha\beta}</math> and a term featuring the damping matrix <math>b_{\alpha\beta}</math> as follows

<center><math>(7)

f_{\beta\alpha}=i\omega(a_{\beta\alpha}+\frac{ib_{\beta\alpha}}{\omega})

</math></center>

where the added mass and damping coefficients are in phase with the acceleration and velocity respectively. These real quantities depend only on the geometry of the structure and the oscillation frequency <math>\omega</math> and describe many of the properties of the structure. In particular, (as will be shown see later) if the variation of these coefficients with frequency is known then we can predict the resonant behaviour of the structure in the time-domain as well as the frequency domain. The computation of these quantities was considered to be very important in marine engineering and naval architecture and so a variety of different computer codes were developed to compute the coefficients efficiently. The boundary element method forms the basis of most of the computer code algorithms, the most widely-used example being WAMIT (see http:<math>\backslash\backslash</math>www.wamit.com) which determines frequency domain solutions for prescribed structures and structure motions.

==Frequency-domain equation of motion==

The radiation force coefficients also play a significant role in determining the motion of a floating structure possibly subject to mooring and applied forces. To obtain the correct equation of motion in the frequency domain it is important to adopt the approach similar to that taken by [[McIver 2005]], i.e. keep the initial condition terms in the Fourier transform operations. Therefore, if the Fourier transform of the velocity is <math>v_{\mu}(\omega)</math> then the Fourier transform of the acceleration is given by

<center><math>

\int_{0}^{\infty}\ddot{X}_{\mu}(t)e^{i\omega t}dt=-i\omega v_{\mu}(\omega) - V_{\mu}(0)

</math></center>

the time-derivative of the potential obeys a similar relation

<center><math>

\int_{0}^{\infty}\frac{\partial\Phi}{\partial t}e^{i\omega t}dt=-i\omega \phi_{\mu}(\omega) - \Phi(\mathbf{x},0).

</math></center>

Therefore, the ini\partial conditions of the structure <math>(X_{\mu}(0),V_{\mu}(0))</math> and of the potential <math>\Phi(\mathbf{x},0)</math> will be present in the Fourier transform of the equation motion which will be referred to as the frequency-domain equation of motion. If the frequency domain potential is decomposed in the same manner as before then the Fourier transform of the equation of motion for the structure is
<center><math>
M\left[-V_{\mu}(0)-i\omega v_{\mu}(\omega)\right]
=-\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS+F^{S}_{\mu}(\omega)+
i\omega \sum_{\nu} (f_{\mu\nu}(\omega)+i\gamma_{\mu\nu}/\omega)v_{\nu}(\omega)-
\sum_{\nu} c_{\mu\nu}\frac{v(\omega)+X(0)}{-i\omega}+f^{A}_{\mu}(\omega)
</math></center>
for <math>\mu=1,\ldots,6</math>, where <math>f^{A}_{\mu}(\omega)</math> is the Fourier transform of the applied force <math>F_{\mu}(t)</math> in equation~(\ref{linearisedmotion}). Although it is assumed that <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math>, for a non-zero initial velocity <math>\lim_{t^{+}\rightarrow 0}\Phi(\mathbf{x},t)\neq 0</math> because of the impulsive pressure on the free-surface resulting from the initial motion of the pressure. Therefore, the first term on the right-hand side of the above equation is not identically zero, rather is given by
<center><math>
\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS=a(\infty)V(0)
</math></center>
[[Mei 1983]]
where <math>a(\infty)=\lim_{\omega\rightarrow\infty}a(\omega)</math> is the infinite frequency added mass.

The frequency-domain equation is usually re-expressed in the following form
<center><math>
\sum_{\nu} \{c_{\mu\nu}-\omega^{2}\left[M_{\mu\mu}\delta_{\mu\nu}+a_{\mu\nu}(\omega)+i(b_{\mu\nu}+\gamma_{\mu\nu})/\omega\right]\}v_{\nu}(\omega)
= -i\omega\left[ F_{\mu}^{S}(\omega)+f_{\mu}(\omega)+(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)\right]-\sum_{\nu}c_{\mu\nu}X_{\nu}(0)
</math></center>
for <math>\mu=1,\ldots,6</math>. If it is assumed that the structure can only move in one mode of motion and that there is no incident wave or applied force then the complex velocity will be given by
<center><math>
v_{\mu}(\omega)=\frac{-i\omega(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)-c_{\mu\mu}X_{\mu}(0)}{c_{\mu\mu}-\omega^{2}\left[M_{\mu\mu}+a_{\mu\mu}(\omega)+i(b_{\mu\mu}+\gamma_{\mu\mu})/\omega\right]}.
</math></center>

As described by [[McIver 2006]], the zero of the denominator gives the location of the pole referred to as the motion resonance. The complex force coefficient <math>f_{\mu\nu}(\omega)</math> will also possess a pole, however this resonance is annulled by the frequency-dependence of the complex velocity (see [[McIver 2006]]) and so the motion, in the absence of incident waves, is dominated by the motion resonance term. Solving for <math>v(\omega)</math> then the time-domain velocity can be recovered using the inverse transform
<center><math>
V(t)=\frac{1}{\pi} Re \int^{\infty}_{0}v(\omega)e^{-i\omega t} dt.
</math></center>

==Integro-differential equation method==

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by <math>\delta(t-\tau)V_{\alpha}(\tau)</math> where <math>\dot{X_{\alpha}}=V_{\alpha}</math>. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function <math>V_{\alpha}(t)</math> an integral of the impulse response over the range <math>-\infty<\tau<\infty</math> must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

<center><math>(11)

\mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau

</math></center>

where <math>m_{\beta\alpha}(\infty)</math> is an infinite frequency added mass coefficient and <math>L_{\beta\alpha}(t-\tau)</math> is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. <math>V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \}</math>, the impulse response function <math>L_{\beta\alpha}</math> can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

<center><math>

\begin{split}

a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, d\tau \

b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, d\tau

\end{split}

</math></center>

and so if the added mass or damping coefficients are known for all frequencies <math>0\leq\omega\leq\infty</math>, the impulse response or memory function <math>L_{\alpha\beta}(t)</math> can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

<center><math>

\left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t),

</math></center>
for <math>\alpha=1,\ldots,6</math>, where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency <math>\omega</math> on the interval <math>[0,\infty)</math>. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

==Integro-differential equation method==

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by <math>\delta(t-\tau)V_{\alpha}(\tau)</math> where <math>\dot{X_{\alpha}}=V_{\alpha}</math>. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function <math>V_{\alpha}(t)</math> an integral of the impulse response over the range <math>-\infty<\tau<\infty</math> must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

<center><math>(11)

\mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau

</math></center>

where <math>m_{\beta\alpha}(\infty)</math> is an infinite frequency added mass coefficient and <math>L_{\beta\alpha}(t-\tau)</math> is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. <math>V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \}</math>, the impulse response function <math>L_{\beta\alpha}</math> can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

<center><math>

\begin{split}

a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, d\tau \

b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, d\tau

\end{split}

</math></center>

and so if the added mass or damping coefficients are known for all frequencies <math>0\leq\omega\leq\infty</math>, the impulse response or memory function <math>L_{\alpha\beta}(t)</math> can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

<center><math>

\left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t),

</math></center>
for <math>\alpha=1,\ldots,6</math>, where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency <math>\omega</math> on the interval <math>[0,\infty)</math>. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

Category:Time-Dependent Linear Water Waves

2009-09-03T01:49:13Z

Cfitzgerald: /* Two dimensional equations for a floating body */

[[Category:Linear Water-Wave Theory]]

{{incomplete pages}}

Generally the focus of research is on the [[Frequency Domain Problem]]. The time-domain
problem can be solved by [[Generalized Eigenfunction Expansion]] or by
an [[Expansion over the Resonances]] or using [[Memory Effect Function]].

== Introduction ==

Analytical solutions of the linearised time-domain equations for a coupled-motion or forced-motion problems are very rare. In fact, only a small number of analytic time-domain solutions have been obtained (see the papers by [[Kennard 1949]] and [[McIver 1994]]) for structures with simple geometries. Therefore, a variety of numerical methods have been developed to solve initial-value time-domain wave-structure interaction problems. The Fourier transform enables the time and frequency domain quantities to be related and the frequency domain forces and potentials have important roles in the development of the numerical time-domain solution methods. This can be partly attributed to the fact that, in the past, analytical or semi-analytical solutions to linearised water-wave problems were more easily obtained in the frequency domain given that the assumption of time-harmonic motion results in a significant simplification of the interaction equations. Therefore, it was natural to develop time-domain solutions based on frequency domain results and the Fourier transform relation between the domains.

== Equations of motion in the Time Domain ==

=== Two Dimensional Equations for fixed bodies in the time domain ===

{{Equations for fixed bodies in the time domain}}

=== Two dimensional equations for a floating body ===
We now consider the equations for a floating two-dimensional structure.
{{equations of motion time domain without body condition}}

The body boundary condition for a fixed body is given in terms
of the 3 rigid body motions, namely surge, heave and pitch which are indexed as <math>\mu=1,3,5</math> in order to be consistent with the three-dimensional problem. We have a kinematic condition
<center><math>
\partial_{n}\Phi=\sum\partial_t \xi_{\nu}\mathbf{n}_{\nu},\ \mathbf{x}\in\partial\Omega,
</math></center>
where <math>\xi_{\nu}</math> is the motion of the <math>\mu</math>th mode and
<math>\mathbf{n}_{\nu}</math> is the normal associated with this mode.
The dynamic condition is the equation of motion for the structure:
<center><math>
M_{\mu\nu}\partial_t^2 \xi_{\nu}=-\rho\iint_{\partial\Omega}\partial_t\Phi n_{\mu}\, dS- c_{\mu\nu}\xi_{\nu},\quad \textrm{for} \qquad \mu=1,3,5,
</math></center>
where summation over repeated indices is assumed. In this equation, <math>M_{\mu\nu}</math> are the elements of the mass matrix
<center><math>
\mathbf{M}=\left[
\begin{matrix}
M & 0 & M(z^c-Z^R) \\
0 & M & -M(x^c-X^R) \\
M(z^c-Z^R)& -M(x^c-X^R) & I^b_{11}+I^b_{33}
\end{matrix}
\right] ,
</math></center>
for the structure and <math>c_{\mu\nu}</math> are the elements of the buoyancy matrix
<center><math>
\mathbf{C}=\left[
\begin{matrix}
0 & 0 & 0 \\
0 & \rho g W & -\rho g I^A_{1} \\
0 & -\rho g I^A_{1} & \begin{matrix}\rho g (I^A_{11}+I^V_3)-\\
Mg(z^c-Z^R) \end{matrix}
\end{matrix}
\right].
</math></center>
The terms <math>I^b_{11}</math>, <math>I^b_{33}</math> are the moments of inertia of the body about the <math>x</math> and <math>z</math> axes and the terms <math>I_1^{A}</math>, <math>I^{A}_{11}</math> are the first and second moments of the waterplane (the waterplane area is denoted <math>W</math>) about the <math>x</math>-axis (see Chapter 7, [[Mei 1983]]). In addition, <math>(x^c,z^c)</math> and <math>(X^R,Z^R)</math> are the positions of the centre of mass and centre of rotation of the body and <math>I^{V}_{3}</math> is <math>z</math>-component centre of buoyancy of the structure. Thus, the coupled equations of motion for a floating structure have been derived. (N.B. if is assumed that the centre of rotation and the centre of mass of the structure coincide, i.e. if it is assumed that the body is semi-submerged, the mass and buoyancy matrices become diagonal). Any wave incidence is assumed to be propagating in the positive <math>x</math> direction.) The scattering and radiation problems are simpler than the coupled problem because the motion of the the structure is then prescribed.

The initial conditions are
{{initial free surface time domain}}
and
{{initial conditions floating body}}

=== Floating body constrained to move in heave ===
The simplest type of floating body problem concerns the motion of a body constrained to move in heave in two-dimensions. Apart from the boundary condition on the structure surface which becomes
<center><math>
\partial_n \Phi = \partial_t\xi_{3}\mathbf{n}_{3},
</math></center>
where <math>\xi_{3}<\math> is the heave displacement of the structure, the equations governing the motion of the fluid remain the same. The equation of motion for a body constrained to move in heave only is
<center><math>
M\partial_t^2 \xi_{3}=-\rho\iint_{\partial\Omega}\partial_t\Phi n_{3}\, dS- \rho g W\xi_{3}.
</math></center>
which is a significant simplification compared to the general case. The mass matrix and buoyancy matrix are replaced by the mass <math>M</math> and the hydrostatic term <math>\rho g W</math>.

==Frequency domain definitions==

Linearised water-wave problems were first investigated in the frequency domain wherein all motions are assumed to be time-harmonic and to have begun from <math>t= -\infty</math> . The time-dependence can be removed in a straightforward manner by substituting for the potential and any structural motions using
<center><math>
\Phi(\mathbf{x},t)=Re \{\phi(\mathbf{x},\omega) e^{-i\omega t}\}
</math></center>
<center><math>
V(\mathbf{x},t)=Re \{ v(\mathbf{x},\omega) e^{-i\omega t}\}
</math></center>
and this results in a considerable simplification of the problem. The complex amplitudes of the dynamic quantities vary with frequency and by removing the time-dependence the governing equations and boundary conditions will feature these terms only.
However, \citeasnoun{mciver2003} have shown that the frequency-domain potential can also be defined using the following operations on the time-domain solution:

*take a Laplace transform of the time-domain potential
<center><math>
\hat{\phi}(\mathbf{x},s)=\int^{\infty}_{0}\Phi(\mathbf{x},t)e^{-s t}\, dt ,\quad \textrm{Re } s >0;
</math></center>

*apply the change of variables <math>s=-i\omega</math> so that
<center><math>
\phi(\mathbf{x},\omega)=\hat{\phi}(\mathbf{x},-i\omega).
</math></center>

*noting that <math>\phi(\mathbf{x},-\omega)=\bar{\phi}(\mathbf{x},\omega)</math>, the inverse Fourier transform is given by
<center><math>(3)
\Phi(\mathbf{x},t)=\frac{1}{2\pi}\int^{\infty}_{-\infty}\phi(\mathbf{x},\omega)e^{-i\omega t} d\omega =\frac{1}{\pi}Re\int^{\infty}_{0}\phi(\mathbf{x},\omega)e^{-i\omega t}\,d\omega
</math></center>
where the path of integration must pass over any singularities of <math>\phi</math> that lie on the real axis and it has been assumed that there is no motion prior to <math>t=0</math>, i.e. <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math>.

If <math>\omega</math> is real then <math>\phi(\mathbf{x},\omega)</math> is just the frequency domain potential. By the principle of causality, which precludes the existence of the response before the cause so that <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math> , no singularities can exist in <math>\textrm{Im } \omega>0</math>. Therefore, if it is assumed that no singularities exist on the real axis the inverse transform will integrate the frequency-domain potential over the full range of <math>\omega</math>. With this approach, the frequency-domain equations are obtained by applying the Fourier transform to the time-dependent equations. However, unlike the equations resulting from the assumptions~(2) of time-harmonic motion from <math>t= -\infty</math>, the initial conditions will be present in the frequency-domain equations arising from the Fourier transformation. Although, the latter equations are more mathematically consistent it is usual to set all initial conditions to zero before attempting to solve the equations even if this is not the case. Nevertheless, in the frequency-domain analysis of resonances in coupled motion problems \citeasnoun{pmciver2005} retains the initial condition terms in the equation of motion for the structure and obtains some important results regarding the resonant behaviour of the motion. Therefore, the retention of the initial condition terms depends on the application of the subsequent equations.

Many analytical and approximate solutions exist for frequency-domain problems involving structures with simple geometries. Thus, in the simplest frequency-domain radiation and diffraction problems it is possible to completely determine the velocity potential analytically. The hydrodynamic forces, which are crucial in the context of marine engineering, can then be calculated directly from the potential. For more general geometries this is not possible and the hydrodynamic forces must be computed using numerical methods. Nevertheless, it is useful to consider the analytical expressions for the various potentials and forces in the frequency-domain problem.

In a linearised coupled motion problem, it is customary to decompose the total velocity potential into a scattering potential <math>\phi^{S}</math> and a radiation potential <math>\phi^{R}</math>. Thus, the coupled motion problem will require the solution of the radiation problem and the scattering problem. The radiation potential can be further decomposed into a sum over the six modes of motion so that the total frequency domain potential is
<center><math>
\phi=\phi^{S}+\sum_{\mu}v_{\mu}\phi_{\mu}
</math></center>

where <math>u_{\mu}</math> is the complex amplitude of the generalised velocity in the <math>\mu</math> direction and <math>\phi_{\mu}</math> describes the fluid response due to the forced oscillations in mode <math>\mu</math> with unit velocity amplitude. The velocities in each mode will be determined from the frequency domain equation of motion for the body. However, this requires the computation of the radiation and exciting forces on the structure. Therefore, it is necessary to consider the radiation problem prior to the coupled motion equation. It is first important to note that, for a given mode <math>\mu</math> the boundary condition on the structure will be

<center><math>

\frac{\partial\phi_{\mu}}{\partial n}=n_{\mu},

</math></center>

because the total velocity is

<center><math>

v(\omega)=\sum_{\mu}v_{\mu}n_{\mu}

</math></center>

where <math>n_{\mu}</math> is the <math>\mu</math> component of the generalised normal and <math>v_{\mu}</math> is the component of the generalised complex velocity amplitude in this direction. The hydrodynamic force on the structure in the <math>\mu</math> direction due to the fluid response to the forced oscillations is

<center><math>(5)

F_{\mu}^{R}=i\omega\rho\iint_{S_{B}}\phi^{R}n_{\mu} \,dS

</math></center>

and the force due to the diffraction of an incident wave by the fixed structure is

<center><math>(6)

F_{\mu}^{S}=i\omega\rho\iint_{S_{B}}\phi^{S}n_{\mu} \,dS

</math></center>

where the time-dependence has been removed in both cases. By expressing the radiation potential as a sum over the individual modes, the radiation force can written as <math>\sum_{\nu} v_{\nu}f_{\nu\mu}</math> where

<center><math>

f_{\nu\mu}=i\omega\rho\iint_{\Gamma} \phi_{\nu}n_{\mu} \,dS.

</math></center>

It is conventional to decompose the radiation force into a term featuring the added mass matrix <math>a_{\alpha\beta}</math> and a term featuring the damping matrix <math>b_{\alpha\beta}</math> as follows

<center><math>(7)

f_{\beta\alpha}=i\omega(a_{\beta\alpha}+\frac{ib_{\beta\alpha}}{\omega})

</math></center>

where the added mass and damping coefficients are in phase with the acceleration and velocity respectively. These real quantities depend only on the geometry of the structure and the oscillation frequency <math>\omega</math> and describe many of the properties of the structure. In particular, (as will be shown see later) if the variation of these coefficients with frequency is known then we can predict the resonant behaviour of the structure in the time-domain as well as the frequency domain. The computation of these quantities was considered to be very important in marine engineering and naval architecture and so a variety of different computer codes were developed to compute the coefficients efficiently. The boundary element method forms the basis of most of the computer code algorithms, the most widely-used example being WAMIT (see http:<math>\backslash\backslash</math>www.wamit.com) which determines frequency domain solutions for prescribed structures and structure motions.

==Frequency-domain equation of motion==

The radiation force coefficients also play a significant role in determining the motion of a floating structure possibly subject to mooring and applied forces. To obtain the correct equation of motion in the frequency domain it is important to adopt the approach similar to that taken by [[McIver 2005]], i.e. keep the initial condition terms in the Fourier transform operations. Therefore, if the Fourier transform of the velocity is <math>v_{\mu}(\omega)</math> then the Fourier transform of the acceleration is given by

<center><math>

\int_{0}^{\infty}\ddot{X}_{\mu}(t)e^{i\omega t}dt=-i\omega v_{\mu}(\omega) - V_{\mu}(0)

</math></center>

the time-derivative of the potential obeys a similar relation

<center><math>

\int_{0}^{\infty}\frac{\partial\Phi}{\partial t}e^{i\omega t}dt=-i\omega \phi_{\mu}(\omega) - \Phi(\mathbf{x},0).

</math></center>

Therefore, the ini\partial conditions of the structure <math>(X_{\mu}(0),V_{\mu}(0))</math> and of the potential <math>\Phi(\mathbf{x},0)</math> will be present in the Fourier transform of the equation motion which will be referred to as the frequency-domain equation of motion. If the frequency domain potential is decomposed in the same manner as before then the Fourier transform of the equation of motion for the structure is
<center><math>
M\left[-V_{\mu}(0)-i\omega v_{\mu}(\omega)\right]
=-\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS+F^{S}_{\mu}(\omega)+
i\omega \sum_{\nu} (f_{\mu\nu}(\omega)+i\gamma_{\mu\nu}/\omega)v_{\nu}(\omega)-
\sum_{\nu} c_{\mu\nu}\frac{v(\omega)+X(0)}{-i\omega}+f^{A}_{\mu}(\omega)
</math></center>
for <math>\mu=1,\ldots,6</math>, where <math>f^{A}_{\mu}(\omega)</math> is the Fourier transform of the applied force <math>F_{\mu}(t)</math> in equation~(\ref{linearisedmotion}). Although it is assumed that <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math>, for a non-zero initial velocity <math>\lim_{t^{+}\rightarrow 0}\Phi(\mathbf{x},t)\neq 0</math> because of the impulsive pressure on the free-surface resulting from the initial motion of the pressure. Therefore, the first term on the right-hand side of the above equation is not identically zero, rather is given by
<center><math>
\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS=a(\infty)V(0)
</math></center>
[[Mei 1983]]
where <math>a(\infty)=\lim_{\omega\rightarrow\infty}a(\omega)</math> is the infinite frequency added mass.

The frequency-domain equation is usually re-expressed in the following form
<center><math>
\sum_{\nu} \{c_{\mu\nu}-\omega^{2}\left[M_{\mu\mu}\delta_{\mu\nu}+a_{\mu\nu}(\omega)+i(b_{\mu\nu}+\gamma_{\mu\nu})/\omega\right]\}v_{\nu}(\omega)
= -i\omega\left[ F_{\mu}^{S}(\omega)+f_{\mu}(\omega)+(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)\right]-\sum_{\nu}c_{\mu\nu}X_{\nu}(0)
</math></center>
for <math>\mu=1,\ldots,6</math>. If it is assumed that the structure can only move in one mode of motion and that there is no incident wave or applied force then the complex velocity will be given by
<center><math>
v_{\mu}(\omega)=\frac{-i\omega(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)-c_{\mu\mu}X_{\mu}(0)}{c_{\mu\mu}-\omega^{2}\left[M_{\mu\mu}+a_{\mu\mu}(\omega)+i(b_{\mu\mu}+\gamma_{\mu\mu})/\omega\right]}.
</math></center>

As described by [[McIver 2006]], the zero of the denominator gives the location of the pole referred to as the motion resonance. The complex force coefficient <math>f_{\mu\nu}(\omega)</math> will also possess a pole, however this resonance is annulled by the frequency-dependence of the complex velocity (see [[McIver 2006]]) and so the motion, in the absence of incident waves, is dominated by the motion resonance term. Solving for <math>v(\omega)</math> then the time-domain velocity can be recovered using the inverse transform
<center><math>
V(t)=\frac{1}{\pi} Re \int^{\infty}_{0}v(\omega)e^{-i\omega t} dt.
</math></center>

==Integro-differential equation method==

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by <math>\delta(t-\tau)V_{\alpha}(\tau)</math> where <math>\dot{X_{\alpha}}=V_{\alpha}</math>. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function <math>V_{\alpha}(t)</math> an integral of the impulse response over the range <math>-\infty<\tau<\infty</math> must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

<center><math>(11)

\mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau

</math></center>

where <math>m_{\beta\alpha}(\infty)</math> is an infinite frequency added mass coefficient and <math>L_{\beta\alpha}(t-\tau)</math> is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. <math>V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \}</math>, the impulse response function <math>L_{\beta\alpha}</math> can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

<center><math>

\begin{split}

a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, d\tau \

b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, d\tau

\end{split}

</math></center>

and so if the added mass or damping coefficients are known for all frequencies <math>0\leq\omega\leq\infty</math>, the impulse response or memory function <math>L_{\alpha\beta}(t)</math> can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

<center><math>

\left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t),

</math></center>
for <math>\alpha=1,\ldots,6</math>, where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency <math>\omega</math> on the interval <math>[0,\infty)</math>. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

==Integro-differential equation method==

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by <math>\delta(t-\tau)V_{\alpha}(\tau)</math> where <math>\dot{X_{\alpha}}=V_{\alpha}</math>. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function <math>V_{\alpha}(t)</math> an integral of the impulse response over the range <math>-\infty<\tau<\infty</math> must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

<center><math>(11)

\mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau

</math></center>

where <math>m_{\beta\alpha}(\infty)</math> is an infinite frequency added mass coefficient and <math>L_{\beta\alpha}(t-\tau)</math> is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. <math>V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \}</math>, the impulse response function <math>L_{\beta\alpha}</math> can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

<center><math>

\begin{split}

a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, d\tau \

b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, d\tau

\end{split}

</math></center>

and so if the added mass or damping coefficients are known for all frequencies <math>0\leq\omega\leq\infty</math>, the impulse response or memory function <math>L_{\alpha\beta}(t)</math> can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

<center><math>

\left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t),

</math></center>
for <math>\alpha=1,\ldots,6</math>, where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency <math>\omega</math> on the interval <math>[0,\infty)</math>. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

Category:Time-Dependent Linear Water Waves

2009-09-02T06:00:01Z

Cfitzgerald: /* Two dimensional equations for a floating body */

[[Category:Linear Water-Wave Theory]]

{{incomplete pages}}

Generally the focus of research is on the [[Frequency Domain Problem]]. The time-domain
problem can be solved by [[Generalized Eigenfunction Expansion]] or by
an [[Expansion over the Resonances]] or using [[Memory Effect Function]].

== Introduction ==

Analytical solutions of the linearised time-domain equations for a coupled-motion or forced-motion problems are very rare. In fact, only a small number of analytic time-domain solutions have been obtained (see the papers by [[Kennard 1949]] and [[McIver 1994]]) for structures with simple geometries. Therefore, a variety of numerical methods have been developed to solve initial-value time-domain wave-structure interaction problems. The Fourier transform enables the time and frequency domain quantities to be related and the frequency domain forces and potentials have important roles in the development of the numerical time-domain solution methods. This can be partly attributed to the fact that, in the past, analytical or semi-analytical solutions to linearised water-wave problems were more easily obtained in the frequency domain given that the assumption of time-harmonic motion results in a significant simplification of the interaction equations. Therefore, it was natural to develop time-domain solutions based on frequency domain results and the Fourier transform relation between the domains.

== Equations of motion in the Time Domain ==

=== Two Dimensional Equations for fixed bodies in the time domain ===

{{Equations for fixed bodies in the time domain}}

=== Two dimensional equations for a floating body ===
We now consider the equations for a floating two-dimensional structure.
{{equations of motion time domain without body condition}}

The body boundary condition for a fixed body is given in terms
of the 3 rigid body motions, namely surge, heave and pitch which are indexed as <math>\mu=1,3,5</math> in order to be consistent with the three-dimensional problem. We have a kinematic condition
<center><math>
\partial_{n}\Phi=\sum\partial_t \eta_{\nu}\mathbf{n}_{\nu},\ \mathbf{x}\in\partial\Omega,
</math></center>
where <math>\eta_{\nu}</math> is the motion of the <math>\mu</math>th mode and
<math>\mathbf{n}_{\nu}</math> is the normal associated with this mode.
The dynamic condition is the equation of motion for the structure:
<center><math>
M_{\mu\nu}\ddot{X}_{\mu}=-\rho\iint_{\Gamma}\frac{\partial\Phi}{\partial t} n_{\mu}\, dS-\sum_{\nu=1,3,5} c_{\mu\nu}X_{\nu},\quad for \qquad \mu=1,3,5,
</math></center>
where <math>M_{\mu\nu}</math> are the elements of the mass matrix
<center><math>
\mathbf{M}=\left[
\begin{matrix}
M & 0 & M(z^c-Z^R) \\
0 & M & -M(x^c-X^R) \\
M(z^c-Z^R)& -M(x^c-X^R) & I^b_{11}+I^b_{33}
\end{matrix}
\right] ,
</math></center>
for the structure and <math>c_{\mu\nu}</math> are the elements of the buoyancy matrix
<center><math>
\mathbf{C}=\left[
\begin{matrix}
0 & 0 & 0 \\
0 & \rho g W & -\rho g I^A_{1} \\
0 & -\rho g I^A_{1} & \rho g (I^A_{11}+I^V_3)-Mg(z^c-Z^R)
\end{matrix}
\right].
</math></center>
The terms <math>I^b_{11}</math>, <math>I^b_{33}</math> are the moments of inertia of the body about the <math>x</math> and <math>z</math> axes and the terms <math>I_1^{A}</math>, <math>I^{A}_{11}</math> are the first and second moments of the waterplane (the waterplane area is denoted <math>W</math>) about the <math>x</math>-axis (see Chapter 7, Mei (1983)). In addition, <math>(x^c,z^c)</math> and <math>(X^R,Z^R)</math> are the positions of the centre of mass and centre of rotation of the body and <math>I^{V}_{3}</math> is <math>z</math>-component centre of buoyancy of the structure. Thus, the coupled equations of motion for a floating structure have been derived. (N.B. if is assumed that the centre of rotation and the centre of mass of the structure coincide, i.e. if it is assumed that the body is semi-submerged, the mass and buoyancy matrices become diagonal). Any wave incidence is assumed to be propagating in the positive <math>x</math> direction.) The scattering and radiation problems are simpler than the coupled problem because the motion of the the structure is then prescribed.
The initial conditions are
{{initial free surface time domain}}
and
{{initial conditions floating body}}

==Frequency domain definitions==

Linearised water-wave problems were first investigated in the frequency domain wherein all motions are assumed to be time-harmonic and to have begun from <math>t= -\infty</math> . The time-dependence can be removed in a straightforward manner by substituting for the potential and any structural motions using
<center><math>
\Phi(\mathbf{x},t)=Re \{\phi(\mathbf{x},\omega) e^{-i\omega t}\}
</math></center>
<center><math>
V(\mathbf{x},t)=Re \{ v(\mathbf{x},\omega) e^{-i\omega t}\}
</math></center>
and this results in a considerable simplification of the problem. The complex amplitudes of the dynamic quantities vary with frequency and by removing the time-dependence the governing equations and boundary conditions will feature these terms only.
However, \citeasnoun{mciver2003} have shown that the frequency-domain potential can also be defined using the following operations on the time-domain solution:

*take a Laplace transform of the time-domain potential
<center><math>
\hat{\phi}(\mathbf{x},s)=\int^{\infty}_{0}\Phi(\mathbf{x},t)e^{-s t}\, dt ,\quad \textrm{Re } s >0;
</math></center>

*apply the change of variables <math>s=-i\omega</math> so that
<center><math>
\phi(\mathbf{x},\omega)=\hat{\phi}(\mathbf{x},-i\omega).
</math></center>

*noting that <math>\phi(\mathbf{x},-\omega)=\bar{\phi}(\mathbf{x},\omega)</math>, the inverse Fourier transform is given by
<center><math>(3)
\Phi(\mathbf{x},t)=\frac{1}{2\pi}\int^{\infty}_{-\infty}\phi(\mathbf{x},\omega)e^{-i\omega t} d\omega =\frac{1}{\pi}Re\int^{\infty}_{0}\phi(\mathbf{x},\omega)e^{-i\omega t}\,d\omega
</math></center>
where the path of integration must pass over any singularities of <math>\phi</math> that lie on the real axis and it has been assumed that there is no motion prior to <math>t=0</math>, i.e. <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math>.

If <math>\omega</math> is real then <math>\phi(\mathbf{x},\omega)</math> is just the frequency domain potential. By the principle of causality, which precludes the existence of the response before the cause so that <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math> , no singularities can exist in <math>\textrm{Im } \omega>0</math>. Therefore, if it is assumed that no singularities exist on the real axis the inverse transform will integrate the frequency-domain potential over the full range of <math>\omega</math>. With this approach, the frequency-domain equations are obtained by applying the Fourier transform to the time-dependent equations. However, unlike the equations resulting from the assumptions~(2) of time-harmonic motion from <math>t= -\infty</math>, the initial conditions will be present in the frequency-domain equations arising from the Fourier transformation. Although, the latter equations are more mathematically consistent it is usual to set all initial conditions to zero before attempting to solve the equations even if this is not the case. Nevertheless, in the frequency-domain analysis of resonances in coupled motion problems \citeasnoun{pmciver2005} retains the initial condition terms in the equation of motion for the structure and obtains some important results regarding the resonant behaviour of the motion. Therefore, the retention of the initial condition terms depends on the application of the subsequent equations.

Many analytical and approximate solutions exist for frequency-domain problems involving structures with simple geometries. Thus, in the simplest frequency-domain radiation and diffraction problems it is possible to completely determine the velocity potential analytically. The hydrodynamic forces, which are crucial in the context of marine engineering, can then be calculated directly from the potential. For more general geometries this is not possible and the hydrodynamic forces must be computed using numerical methods. Nevertheless, it is useful to consider the analytical expressions for the various potentials and forces in the frequency-domain problem.

In a linearised coupled motion problem, it is customary to decompose the total velocity potential into a scattering potential <math>\phi^{S}</math> and a radiation potential <math>\phi^{R}</math>. Thus, the coupled motion problem will require the solution of the radiation problem and the scattering problem. The radiation potential can be further decomposed into a sum over the six modes of motion so that the total frequency domain potential is
<center><math>
\phi=\phi^{S}+\sum_{\mu}v_{\mu}\phi_{\mu}
</math></center>

where <math>u_{\mu}</math> is the complex amplitude of the generalised velocity in the <math>\mu</math> direction and <math>\phi_{\mu}</math> describes the fluid response due to the forced oscillations in mode <math>\mu</math> with unit velocity amplitude. The velocities in each mode will be determined from the frequency domain equation of motion for the body. However, this requires the computation of the radiation and exciting forces on the structure. Therefore, it is necessary to consider the radiation problem prior to the coupled motion equation. It is first important to note that, for a given mode <math>\mu</math> the boundary condition on the structure will be

<center><math>

\frac{\partial\phi_{\mu}}{\partial n}=n_{\mu},

</math></center>

because the total velocity is

<center><math>

v(\omega)=\sum_{\mu}v_{\mu}n_{\mu}

</math></center>

where <math>n_{\mu}</math> is the <math>\mu</math> component of the generalised normal and <math>v_{\mu}</math> is the component of the generalised complex velocity amplitude in this direction. The hydrodynamic force on the structure in the <math>\mu</math> direction due to the fluid response to the forced oscillations is

<center><math>(5)

F_{\mu}^{R}=i\omega\rho\iint_{S_{B}}\phi^{R}n_{\mu} \,dS

</math></center>

and the force due to the diffraction of an incident wave by the fixed structure is

<center><math>(6)

F_{\mu}^{S}=i\omega\rho\iint_{S_{B}}\phi^{S}n_{\mu} \,dS

</math></center>

where the time-dependence has been removed in both cases. By expressing the radiation potential as a sum over the individual modes, the radiation force can written as <math>\sum_{\nu} v_{\nu}f_{\nu\mu}</math> where

<center><math>

f_{\nu\mu}=i\omega\rho\iint_{\Gamma} \phi_{\nu}n_{\mu} \,dS.

</math></center>

It is conventional to decompose the radiation force into a term featuring the added mass matrix <math>a_{\alpha\beta}</math> and a term featuring the damping matrix <math>b_{\alpha\beta}</math> as follows

<center><math>(7)

f_{\beta\alpha}=i\omega(a_{\beta\alpha}+\frac{ib_{\beta\alpha}}{\omega})

</math></center>

where the added mass and damping coefficients are in phase with the acceleration and velocity respectively. These real quantities depend only on the geometry of the structure and the oscillation frequency <math>\omega</math> and describe many of the properties of the structure. In particular, (as will be shown see later) if the variation of these coefficients with frequency is known then we can predict the resonant behaviour of the structure in the time-domain as well as the frequency domain. The computation of these quantities was considered to be very important in marine engineering and naval architecture and so a variety of different computer codes were developed to compute the coefficients efficiently. The boundary element method forms the basis of most of the computer code algorithms, the most widely-used example being WAMIT (see http:<math>\backslash\backslash</math>www.wamit.com) which determines frequency domain solutions for prescribed structures and structure motions.

==Frequency-domain equation of motion==

The radiation force coefficients also play a significant role in determining the motion of a floating structure possibly subject to mooring and applied forces. To obtain the correct equation of motion in the frequency domain it is important to adopt the approach similar to that taken by [[McIver 2005]], i.e. keep the initial condition terms in the Fourier transform operations. Therefore, if the Fourier transform of the velocity is <math>v_{\mu}(\omega)</math> then the Fourier transform of the acceleration is given by

<center><math>

\int_{0}^{\infty}\ddot{X}_{\mu}(t)e^{i\omega t}dt=-i\omega v_{\mu}(\omega) - V_{\mu}(0)

</math></center>

the time-derivative of the potential obeys a similar relation

<center><math>

\int_{0}^{\infty}\frac{\partial\Phi}{\partial t}e^{i\omega t}dt=-i\omega \phi_{\mu}(\omega) - \Phi(\mathbf{x},0).

</math></center>

Therefore, the ini\partial conditions of the structure <math>(X_{\mu}(0),V_{\mu}(0))</math> and of the potential <math>\Phi(\mathbf{x},0)</math> will be present in the Fourier transform of the equation motion which will be referred to as the frequency-domain equation of motion. If the frequency domain potential is decomposed in the same manner as before then the Fourier transform of the equation of motion for the structure is
<center><math>
M\left[-V_{\mu}(0)-i\omega v_{\mu}(\omega)\right]
=-\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS+F^{S}_{\mu}(\omega)+
i\omega \sum_{\nu} (f_{\mu\nu}(\omega)+i\gamma_{\mu\nu}/\omega)v_{\nu}(\omega)-
\sum_{\nu} c_{\mu\nu}\frac{v(\omega)+X(0)}{-i\omega}+f^{A}_{\mu}(\omega)
</math></center>
for <math>\mu=1,\ldots,6</math>, where <math>f^{A}_{\mu}(\omega)</math> is the Fourier transform of the applied force <math>F_{\mu}(t)</math> in equation~(\ref{linearisedmotion}). Although it is assumed that <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math>, for a non-zero initial velocity <math>\lim_{t^{+}\rightarrow 0}\Phi(\mathbf{x},t)\neq 0</math> because of the impulsive pressure on the free-surface resulting from the initial motion of the pressure. Therefore, the first term on the right-hand side of the above equation is not identically zero, rather is given by
<center><math>
\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS=a(\infty)V(0)
</math></center>
[[Mei 1983]]
where <math>a(\infty)=\lim_{\omega\rightarrow\infty}a(\omega)</math> is the infinite frequency added mass.

The frequency-domain equation is usually re-expressed in the following form
<center><math>
\sum_{\nu} \{c_{\mu\nu}-\omega^{2}\left[M_{\mu\mu}\delta_{\mu\nu}+a_{\mu\nu}(\omega)+i(b_{\mu\nu}+\gamma_{\mu\nu})/\omega\right]\}v_{\nu}(\omega)
= -i\omega\left[ F_{\mu}^{S}(\omega)+f_{\mu}(\omega)+(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)\right]-\sum_{\nu}c_{\mu\nu}X_{\nu}(0)
</math></center>
for <math>\mu=1,\ldots,6</math>. If it is assumed that the structure can only move in one mode of motion and that there is no incident wave or applied force then the complex velocity will be given by
<center><math>
v_{\mu}(\omega)=\frac{-i\omega(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)-c_{\mu\mu}X_{\mu}(0)}{c_{\mu\mu}-\omega^{2}\left[M_{\mu\mu}+a_{\mu\mu}(\omega)+i(b_{\mu\mu}+\gamma_{\mu\mu})/\omega\right]}.
</math></center>

As described by [[McIver 2006]], the zero of the denominator gives the location of the pole referred to as the motion resonance. The complex force coefficient <math>f_{\mu\nu}(\omega)</math> will also possess a pole, however this resonance is annulled by the frequency-dependence of the complex velocity (see [[McIver 2006]]) and so the motion, in the absence of incident waves, is dominated by the motion resonance term. Solving for <math>v(\omega)</math> then the time-domain velocity can be recovered using the inverse transform
<center><math>
V(t)=\frac{1}{\pi} Re \int^{\infty}_{0}v(\omega)e^{-i\omega t} dt.
</math></center>

==Integro-differential equation method==

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by <math>\delta(t-\tau)V_{\alpha}(\tau)</math> where <math>\dot{X_{\alpha}}=V_{\alpha}</math>. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function <math>V_{\alpha}(t)</math> an integral of the impulse response over the range <math>-\infty<\tau<\infty</math> must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

<center><math>(11)

\mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau

</math></center>

where <math>m_{\beta\alpha}(\infty)</math> is an infinite frequency added mass coefficient and <math>L_{\beta\alpha}(t-\tau)</math> is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. <math>V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \}</math>, the impulse response function <math>L_{\beta\alpha}</math> can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

<center><math>

\begin{split}

a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, d\tau \

b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, d\tau

\end{split}

</math></center>

and so if the added mass or damping coefficients are known for all frequencies <math>0\leq\omega\leq\infty</math>, the impulse response or memory function <math>L_{\alpha\beta}(t)</math> can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

<center><math>

\left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t),

</math></center>
for <math>\alpha=1,\ldots,6</math>, where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency <math>\omega</math> on the interval <math>[0,\infty)</math>. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

==Integro-differential equation method==

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by <math>\delta(t-\tau)V_{\alpha}(\tau)</math> where <math>\dot{X_{\alpha}}=V_{\alpha}</math>. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function <math>V_{\alpha}(t)</math> an integral of the impulse response over the range <math>-\infty<\tau<\infty</math> must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

<center><math>(11)

\mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau

</math></center>

where <math>m_{\beta\alpha}(\infty)</math> is an infinite frequency added mass coefficient and <math>L_{\beta\alpha}(t-\tau)</math> is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. <math>V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \}</math>, the impulse response function <math>L_{\beta\alpha}</math> can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

<center><math>

\begin{split}

a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, d\tau \

b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, d\tau

\end{split}

</math></center>

and so if the added mass or damping coefficients are known for all frequencies <math>0\leq\omega\leq\infty</math>, the impulse response or memory function <math>L_{\alpha\beta}(t)</math> can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

<center><math>

\left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t),

</math></center>
for <math>\alpha=1,\ldots,6</math>, where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency <math>\omega</math> on the interval <math>[0,\infty)</math>. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

==Generalised eigenfunction expansion method==

The singularity expansion method (SEM) is based on a general scattering theory for general wave problems and has been adapted by \citeasnoun{hazardSEM} and \citeasnoun{meylaniwwwfb2008}

to problems in hydrodynamics and hydro-elasticity. The central concept in the theory is to express the governing equations for the scattering problem in terms of an evolution equation involving a unitary operator and to use the generalised eigenfunctions and eigenfrequencies of this operator to give an approximation to the time-domain solution of the scattering problem. To give a brief overview of the method the scattering equations must be re-expressed using a harmonic lifting operator <math>\mbf{B}</math> which maps a function defined on the free-surface onto the total fluid domain. This harmonic lifting operator is denoted by <math>\mbf{B}</math> so that <math>\mbf{B}\Psi=\Phi</math> where <math>\Psi=\Phi</math> on the free-surface <math>F</math>. If <math>D</math> denotes the total fluid domain, then potential <math>\Phi</math> satisfies

<center><math>\begin{matrix}

\nabla^{2}\Phi&=0 \quad\textrm{in } D, \

\frac{tial\Phi}{tial n} &= 0 \quad\textrm{on } tial D,\

\Phi&=\Psi \quad\textrm{on } F

\end{matrix}</math></center>

where <math>tial D</math> includes the bed and the structure surface. The Dirichlet to Neumann map <math>tial_{\mbf{n}}\mbf{B}</math> defined by

<center><math>(12)

tial_{\mbf{n}}\mbf{B}\Psi=\frac{tial\Phi}{tial z}, \quad x\in F,

</math></center>

then recovers the corresponding normal derivative to <math>\Psi</math> on the free surface.

The evolution equation

<center><math>

i\frac{tial \mbf{U}}{tial t} = \mathcal{A}\mbf{U}

</math></center>

accounts for the linear time-dependent free-surface equations using the vector

<center><math>

\mbf{U}= \begin{pmatrix}

\tab \Psi \

\tab -i\eta

\end{pmatrix},

</math></center>

with <math>\eta</math> denoting the time-dependent free-surface elevation, and the operator <math>\mathcal{A}</math> which is both unitary and self-adjoint given by

<center><math>

Category:Time-Dependent Linear Water Waves

2009-09-02T05:58:56Z

Cfitzgerald: /* Two dimensional equations for a floating body */

[[Category:Linear Water-Wave Theory]]

{{incomplete pages}}

Generally the focus of research is on the [[Frequency Domain Problem]]. The time-domain
problem can be solved by [[Generalized Eigenfunction Expansion]] or by
an [[Expansion over the Resonances]] or using [[Memory Effect Function]].

== Introduction ==

Analytical solutions of the linearised time-domain equations for a coupled-motion or forced-motion problems are very rare. In fact, only a small number of analytic time-domain solutions have been obtained (see the papers by [[Kennard 1949]] and [[McIver 1994]]) for structures with simple geometries. Therefore, a variety of numerical methods have been developed to solve initial-value time-domain wave-structure interaction problems. The Fourier transform enables the time and frequency domain quantities to be related and the frequency domain forces and potentials have important roles in the development of the numerical time-domain solution methods. This can be partly attributed to the fact that, in the past, analytical or semi-analytical solutions to linearised water-wave problems were more easily obtained in the frequency domain given that the assumption of time-harmonic motion results in a significant simplification of the interaction equations. Therefore, it was natural to develop time-domain solutions based on frequency domain results and the Fourier transform relation between the domains.

== Equations of motion in the Time Domain ==

=== Two Dimensional Equations for fixed bodies in the time domain ===

{{Equations for fixed bodies in the time domain}}

=== Two dimensional equations for a floating body ===
We now consider the equations for a floating two-dimensional structure.
{{equations of motion time domain without body condition}}

The body boundary condition for a fixed body is given in terms
of the 3 rigid body motions, namely surge, heave and pitch which are indexed as <math>\mu=1,3,5</math> in accordance with the three-dimensional problem. We have a kinematic condition
<center><math>
\partial_{n}\Phi=\sum\partial_t \eta_{\nu}\mathbf{n}_{\nu},\ \mathbf{x}\in\partial\Omega,
</math></center>
where <math>\eta_{\nu}</math> is the motion of the <math>\mu</math>th mode and
<math>\mathbf{n}_{\nu}</math> is the normal associated with this mode.
The dynamic condition is the equation of motion for the structure:
<center><math>
M_{\mu\nu}\ddot{X}_{\mu}=-\rho\iint_{\Gamma}\frac{\partial\Phi}{\partial t} n_{\mu}\, dS-\sum_{\nu=1,3,5} c_{\mu\nu}X_{\nu},\quad for \qquad \mu=1,3,5,
</math></center>
where <math>M_{\mu\nu}</math> are the elements of the mass matrix
<center><math>
\mathbf{M}=\left[
\begin{matrix}
M & 0 & M(z^c-Z^R) \\
0 & M & -M(x^c-X^R) \\
M(z^c-Z^R)& -M(x^c-X^R) & I^b_{11}+I^b_{33}
\end{matrix}
\right] ,
</math></center>
for the structure and <math>c_{\mu\nu}</math> are the elements of the buoyancy matrix
<center><math>
\mathbf{C}=\left[
\begin{matrix}
0 & 0 & 0 \\
0 & \rho g W & -\rho g I^A_{1} \\
0 & -\rho g I^A_{1} & \rho g (I^A_{11}+I^V_3)-Mg(z^c-Z^R)
\end{matrix}
\right].
</math></center>
The terms <math>I^b_{11}</math>, <math>I^b_{33}</math> are the moments of inertia of the body about the <math>x</math> and <math>z</math> axes and the terms <math>I_1^{A}</math>, <math>I^{A}_{11}</math> are the first and second moments of the waterplane (the waterplane area is denoted <math>W</math>) about the <math>x</math>-axis (see Chapter 7, Mei (1983)). In addition, <math>(x^c,z^c)</math> and <math>(X^R,Z^R)</math> are the positions of the centre of mass and centre of rotation of the body and <math>I^{V}_{3}</math> is <math>z</math>-component centre of buoyancy of the structure. Thus, the coupled equations of motion for a floating structure have been derived. (N.B. if is assumed that the centre of rotation and the centre of mass of the structure coincide, i.e. if it is assumed that the body is semi-submerged, the mass and buoyancy matrices become diagonal). Any wave incidence is assumed to be propagating in the positive <math>x</math> direction.) The scattering and radiation problems are simpler than the coupled problem because the motion of the the structure is then prescribed.
The initial conditions are
{{initial free surface time domain}}
and
{{initial conditions floating body}}

==Frequency domain definitions==

Linearised water-wave problems were first investigated in the frequency domain wherein all motions are assumed to be time-harmonic and to have begun from <math>t= -\infty</math> . The time-dependence can be removed in a straightforward manner by substituting for the potential and any structural motions using
<center><math>
\Phi(\mathbf{x},t)=Re \{\phi(\mathbf{x},\omega) e^{-i\omega t}\}
</math></center>
<center><math>
V(\mathbf{x},t)=Re \{ v(\mathbf{x},\omega) e^{-i\omega t}\}
</math></center>
and this results in a considerable simplification of the problem. The complex amplitudes of the dynamic quantities vary with frequency and by removing the time-dependence the governing equations and boundary conditions will feature these terms only.
However, \citeasnoun{mciver2003} have shown that the frequency-domain potential can also be defined using the following operations on the time-domain solution:

*take a Laplace transform of the time-domain potential
<center><math>
\hat{\phi}(\mathbf{x},s)=\int^{\infty}_{0}\Phi(\mathbf{x},t)e^{-s t}\, dt ,\quad \textrm{Re } s >0;
</math></center>

*apply the change of variables <math>s=-i\omega</math> so that
<center><math>
\phi(\mathbf{x},\omega)=\hat{\phi}(\mathbf{x},-i\omega).
</math></center>

*noting that <math>\phi(\mathbf{x},-\omega)=\bar{\phi}(\mathbf{x},\omega)</math>, the inverse Fourier transform is given by
<center><math>(3)
\Phi(\mathbf{x},t)=\frac{1}{2\pi}\int^{\infty}_{-\infty}\phi(\mathbf{x},\omega)e^{-i\omega t} d\omega =\frac{1}{\pi}Re\int^{\infty}_{0}\phi(\mathbf{x},\omega)e^{-i\omega t}\,d\omega
</math></center>
where the path of integration must pass over any singularities of <math>\phi</math> that lie on the real axis and it has been assumed that there is no motion prior to <math>t=0</math>, i.e. <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math>.

If <math>\omega</math> is real then <math>\phi(\mathbf{x},\omega)</math> is just the frequency domain potential. By the principle of causality, which precludes the existence of the response before the cause so that <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math> , no singularities can exist in <math>\textrm{Im } \omega>0</math>. Therefore, if it is assumed that no singularities exist on the real axis the inverse transform will integrate the frequency-domain potential over the full range of <math>\omega</math>. With this approach, the frequency-domain equations are obtained by applying the Fourier transform to the time-dependent equations. However, unlike the equations resulting from the assumptions~(2) of time-harmonic motion from <math>t= -\infty</math>, the initial conditions will be present in the frequency-domain equations arising from the Fourier transformation. Although, the latter equations are more mathematically consistent it is usual to set all initial conditions to zero before attempting to solve the equations even if this is not the case. Nevertheless, in the frequency-domain analysis of resonances in coupled motion problems \citeasnoun{pmciver2005} retains the initial condition terms in the equation of motion for the structure and obtains some important results regarding the resonant behaviour of the motion. Therefore, the retention of the initial condition terms depends on the application of the subsequent equations.

Many analytical and approximate solutions exist for frequency-domain problems involving structures with simple geometries. Thus, in the simplest frequency-domain radiation and diffraction problems it is possible to completely determine the velocity potential analytically. The hydrodynamic forces, which are crucial in the context of marine engineering, can then be calculated directly from the potential. For more general geometries this is not possible and the hydrodynamic forces must be computed using numerical methods. Nevertheless, it is useful to consider the analytical expressions for the various potentials and forces in the frequency-domain problem.

In a linearised coupled motion problem, it is customary to decompose the total velocity potential into a scattering potential <math>\phi^{S}</math> and a radiation potential <math>\phi^{R}</math>. Thus, the coupled motion problem will require the solution of the radiation problem and the scattering problem. The radiation potential can be further decomposed into a sum over the six modes of motion so that the total frequency domain potential is
<center><math>
\phi=\phi^{S}+\sum_{\mu}v_{\mu}\phi_{\mu}
</math></center>

where <math>u_{\mu}</math> is the complex amplitude of the generalised velocity in the <math>\mu</math> direction and <math>\phi_{\mu}</math> describes the fluid response due to the forced oscillations in mode <math>\mu</math> with unit velocity amplitude. The velocities in each mode will be determined from the frequency domain equation of motion for the body. However, this requires the computation of the radiation and exciting forces on the structure. Therefore, it is necessary to consider the radiation problem prior to the coupled motion equation. It is first important to note that, for a given mode <math>\mu</math> the boundary condition on the structure will be

<center><math>

\frac{\partial\phi_{\mu}}{\partial n}=n_{\mu},

</math></center>

because the total velocity is

<center><math>

v(\omega)=\sum_{\mu}v_{\mu}n_{\mu}

</math></center>

where <math>n_{\mu}</math> is the <math>\mu</math> component of the generalised normal and <math>v_{\mu}</math> is the component of the generalised complex velocity amplitude in this direction. The hydrodynamic force on the structure in the <math>\mu</math> direction due to the fluid response to the forced oscillations is

<center><math>(5)

F_{\mu}^{R}=i\omega\rho\iint_{S_{B}}\phi^{R}n_{\mu} \,dS

</math></center>

and the force due to the diffraction of an incident wave by the fixed structure is

<center><math>(6)

F_{\mu}^{S}=i\omega\rho\iint_{S_{B}}\phi^{S}n_{\mu} \,dS

</math></center>

where the time-dependence has been removed in both cases. By expressing the radiation potential as a sum over the individual modes, the radiation force can written as <math>\sum_{\nu} v_{\nu}f_{\nu\mu}</math> where

<center><math>

f_{\nu\mu}=i\omega\rho\iint_{\Gamma} \phi_{\nu}n_{\mu} \,dS.

</math></center>

It is conventional to decompose the radiation force into a term featuring the added mass matrix <math>a_{\alpha\beta}</math> and a term featuring the damping matrix <math>b_{\alpha\beta}</math> as follows

<center><math>(7)

f_{\beta\alpha}=i\omega(a_{\beta\alpha}+\frac{ib_{\beta\alpha}}{\omega})

</math></center>

where the added mass and damping coefficients are in phase with the acceleration and velocity respectively. These real quantities depend only on the geometry of the structure and the oscillation frequency <math>\omega</math> and describe many of the properties of the structure. In particular, (as will be shown see later) if the variation of these coefficients with frequency is known then we can predict the resonant behaviour of the structure in the time-domain as well as the frequency domain. The computation of these quantities was considered to be very important in marine engineering and naval architecture and so a variety of different computer codes were developed to compute the coefficients efficiently. The boundary element method forms the basis of most of the computer code algorithms, the most widely-used example being WAMIT (see http:<math>\backslash\backslash</math>www.wamit.com) which determines frequency domain solutions for prescribed structures and structure motions.

==Frequency-domain equation of motion==

The radiation force coefficients also play a significant role in determining the motion of a floating structure possibly subject to mooring and applied forces. To obtain the correct equation of motion in the frequency domain it is important to adopt the approach similar to that taken by [[McIver 2005]], i.e. keep the initial condition terms in the Fourier transform operations. Therefore, if the Fourier transform of the velocity is <math>v_{\mu}(\omega)</math> then the Fourier transform of the acceleration is given by

<center><math>

\int_{0}^{\infty}\ddot{X}_{\mu}(t)e^{i\omega t}dt=-i\omega v_{\mu}(\omega) - V_{\mu}(0)

</math></center>

the time-derivative of the potential obeys a similar relation

<center><math>

\int_{0}^{\infty}\frac{\partial\Phi}{\partial t}e^{i\omega t}dt=-i\omega \phi_{\mu}(\omega) - \Phi(\mathbf{x},0).

</math></center>

Therefore, the ini\partial conditions of the structure <math>(X_{\mu}(0),V_{\mu}(0))</math> and of the potential <math>\Phi(\mathbf{x},0)</math> will be present in the Fourier transform of the equation motion which will be referred to as the frequency-domain equation of motion. If the frequency domain potential is decomposed in the same manner as before then the Fourier transform of the equation of motion for the structure is
<center><math>
M\left[-V_{\mu}(0)-i\omega v_{\mu}(\omega)\right]
=-\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS+F^{S}_{\mu}(\omega)+
i\omega \sum_{\nu} (f_{\mu\nu}(\omega)+i\gamma_{\mu\nu}/\omega)v_{\nu}(\omega)-
\sum_{\nu} c_{\mu\nu}\frac{v(\omega)+X(0)}{-i\omega}+f^{A}_{\mu}(\omega)
</math></center>
for <math>\mu=1,\ldots,6</math>, where <math>f^{A}_{\mu}(\omega)</math> is the Fourier transform of the applied force <math>F_{\mu}(t)</math> in equation~(\ref{linearisedmotion}). Although it is assumed that <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math>, for a non-zero initial velocity <math>\lim_{t^{+}\rightarrow 0}\Phi(\mathbf{x},t)\neq 0</math> because of the impulsive pressure on the free-surface resulting from the initial motion of the pressure. Therefore, the first term on the right-hand side of the above equation is not identically zero, rather is given by
<center><math>
\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS=a(\infty)V(0)
</math></center>
[[Mei 1983]]
where <math>a(\infty)=\lim_{\omega\rightarrow\infty}a(\omega)</math> is the infinite frequency added mass.

The frequency-domain equation is usually re-expressed in the following form
<center><math>
\sum_{\nu} \{c_{\mu\nu}-\omega^{2}\left[M_{\mu\mu}\delta_{\mu\nu}+a_{\mu\nu}(\omega)+i(b_{\mu\nu}+\gamma_{\mu\nu})/\omega\right]\}v_{\nu}(\omega)
= -i\omega\left[ F_{\mu}^{S}(\omega)+f_{\mu}(\omega)+(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)\right]-\sum_{\nu}c_{\mu\nu}X_{\nu}(0)
</math></center>
for <math>\mu=1,\ldots,6</math>. If it is assumed that the structure can only move in one mode of motion and that there is no incident wave or applied force then the complex velocity will be given by
<center><math>
v_{\mu}(\omega)=\frac{-i\omega(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)-c_{\mu\mu}X_{\mu}(0)}{c_{\mu\mu}-\omega^{2}\left[M_{\mu\mu}+a_{\mu\mu}(\omega)+i(b_{\mu\mu}+\gamma_{\mu\mu})/\omega\right]}.
</math></center>

As described by [[McIver 2006]], the zero of the denominator gives the location of the pole referred to as the motion resonance. The complex force coefficient <math>f_{\mu\nu}(\omega)</math> will also possess a pole, however this resonance is annulled by the frequency-dependence of the complex velocity (see [[McIver 2006]]) and so the motion, in the absence of incident waves, is dominated by the motion resonance term. Solving for <math>v(\omega)</math> then the time-domain velocity can be recovered using the inverse transform
<center><math>
V(t)=\frac{1}{\pi} Re \int^{\infty}_{0}v(\omega)e^{-i\omega t} dt.
</math></center>

==Integro-differential equation method==

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by <math>\delta(t-\tau)V_{\alpha}(\tau)</math> where <math>\dot{X_{\alpha}}=V_{\alpha}</math>. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function <math>V_{\alpha}(t)</math> an integral of the impulse response over the range <math>-\infty<\tau<\infty</math> must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

<center><math>(11)

\mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau

</math></center>

where <math>m_{\beta\alpha}(\infty)</math> is an infinite frequency added mass coefficient and <math>L_{\beta\alpha}(t-\tau)</math> is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. <math>V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \}</math>, the impulse response function <math>L_{\beta\alpha}</math> can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

<center><math>

\begin{split}

a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, d\tau \

b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, d\tau

\end{split}

</math></center>

and so if the added mass or damping coefficients are known for all frequencies <math>0\leq\omega\leq\infty</math>, the impulse response or memory function <math>L_{\alpha\beta}(t)</math> can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

<center><math>

\left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t),

</math></center>
for <math>\alpha=1,\ldots,6</math>, where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency <math>\omega</math> on the interval <math>[0,\infty)</math>. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

==Integro-differential equation method==

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by <math>\delta(t-\tau)V_{\alpha}(\tau)</math> where <math>\dot{X_{\alpha}}=V_{\alpha}</math>. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function <math>V_{\alpha}(t)</math> an integral of the impulse response over the range <math>-\infty<\tau<\infty</math> must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

<center><math>(11)

\mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau

</math></center>

where <math>m_{\beta\alpha}(\infty)</math> is an infinite frequency added mass coefficient and <math>L_{\beta\alpha}(t-\tau)</math> is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. <math>V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \}</math>, the impulse response function <math>L_{\beta\alpha}</math> can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

<center><math>

\begin{split}

a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, d\tau \

b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, d\tau

\end{split}

</math></center>

and so if the added mass or damping coefficients are known for all frequencies <math>0\leq\omega\leq\infty</math>, the impulse response or memory function <math>L_{\alpha\beta}(t)</math> can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

<center><math>

\left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t),

</math></center>
for <math>\alpha=1,\ldots,6</math>, where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency <math>\omega</math> on the interval <math>[0,\infty)</math>. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

==Generalised eigenfunction expansion method==

The singularity expansion method (SEM) is based on a general scattering theory for general wave problems and has been adapted by \citeasnoun{hazardSEM} and \citeasnoun{meylaniwwwfb2008}

to problems in hydrodynamics and hydro-elasticity. The central concept in the theory is to express the governing equations for the scattering problem in terms of an evolution equation involving a unitary operator and to use the generalised eigenfunctions and eigenfrequencies of this operator to give an approximation to the time-domain solution of the scattering problem. To give a brief overview of the method the scattering equations must be re-expressed using a harmonic lifting operator <math>\mbf{B}</math> which maps a function defined on the free-surface onto the total fluid domain. This harmonic lifting operator is denoted by <math>\mbf{B}</math> so that <math>\mbf{B}\Psi=\Phi</math> where <math>\Psi=\Phi</math> on the free-surface <math>F</math>. If <math>D</math> denotes the total fluid domain, then potential <math>\Phi</math> satisfies

<center><math>\begin{matrix}

\nabla^{2}\Phi&=0 \quad\textrm{in } D, \

\frac{tial\Phi}{tial n} &= 0 \quad\textrm{on } tial D,\

\Phi&=\Psi \quad\textrm{on } F

\end{matrix}</math></center>

where <math>tial D</math> includes the bed and the structure surface. The Dirichlet to Neumann map <math>tial_{\mbf{n}}\mbf{B}</math> defined by

<center><math>(12)

tial_{\mbf{n}}\mbf{B}\Psi=\frac{tial\Phi}{tial z}, \quad x\in F,

</math></center>

then recovers the corresponding normal derivative to <math>\Psi</math> on the free surface.

The evolution equation

<center><math>

i\frac{tial \mbf{U}}{tial t} = \mathcal{A}\mbf{U}

</math></center>

accounts for the linear time-dependent free-surface equations using the vector

<center><math>

\mbf{U}= \begin{pmatrix}

\tab \Psi \

\tab -i\eta

\end{pmatrix},

</math></center>

with <math>\eta</math> denoting the time-dependent free-surface elevation, and the operator <math>\mathcal{A}</math> which is both unitary and self-adjoint given by

<center><math>

Category:Time-Dependent Linear Water Waves

2009-08-23T23:29:34Z

Cfitzgerald: /* Two dimensional equations for a floating body */

Generally the focus of research is on the [[Frequency Domain Problem]]. The time-domain
problem can be solved by [[Generalised Eigenfunction Expansion]] or by
an [[Expansion over the Resonances]] or using [[Memory Effect Function]].

== Introduction ==

Analytical solutions of the linearised time-domain equations for a coupled-motion or forced-motion problems are very rare. In fact, only a small number of analytic time-domain solutions have been obtained (see the papers by [[Kennard 1949]] and [[McIver 1994]]) for structures with simple geometries. Therefore, a variety of numerical methods have been developed to solve initial-value time-domain wave-structure interaction problems. The Fourier transform enables the time and frequency domain quantities to be related and the frequency domain forces and potentials have important roles in the development of the numerical time-domain solution methods. This can be partly attributed to the fact that, in the past, analytical or semi-analytical solutions to linearised water-wave problems were more easily obtained in the frequency domain given that the assumption of time-harmonic motion results in a significant simplification of the interaction equations. Therefore, it was natural to develop time-domain solutions based on frequency domain results and the Fourier transform relation between the domains.

== Two Dimensional Equations for fixed bodies in the time domain ==

{{Equations for fixed bodies in the time domain}}

== Two dimensional equations for a floating body ==
We now consider the equations for a floating structure.
{{equations of motion time domain without body condition}}

The body boundary condition for a fixed body is given in terms
of the 3 rigid body motions. We have a dynamic condition
<center><math>
\partial_{n}\Phi=\sum\partial_t \eta_{\nu}\mathbf{n}_{\nu},\ \mathbf{x}\in\partial\Omega,
</math></center>
where <math>\eta_{\nu}</math> is the motion of the <math>\mu</math>th mode and
<math>\mathbf{n}_{\nu}</math> is the normal associate with this mode.
The kinematic condition
<center><math>
M_{\mu\mu}\ddot{X}_{\mu}=-\rho\iint_{\Gamma} \frac{\partial\Phi}{\partial t} n_{\mu}\, dS-\sum^{6}_{\nu=1}\left[ c_{\mu\nu}X_{\nu}+\gamma_{\mu\nu}\dot{X}_{\nu}\right]+F_{\mu}(t), \qquad \mu=1,\ldots,6
</math></center>
where <math>M_{\mu\mu}</math> are the diagonal elements of mass matrix for the structure, i.e. <math>M_{\mu\mu}=M</math> for <math>\mu=1,2,3</math> and <math>M_{\mu\mu}=I_{\mu-3,\mu-3}</math>
<math>c_{\mu\nu}=\rho g b_{\mu}\delta_{\mu\nu}+k_{\mu\nu}</math> which also describes the effect of buoyancy.
It is non-zero for the heave, roll and pitch modes only and these terms are <math>b_{3}=W</math>, <math>b_{4}=I^{A}_{22}+I^{V}_{3}</math> and <math>b_{5}=I^{A}_{11}+I^{V}_{3}</math> respectively, where <math>W</math> is the waterplane area and <math>I^{A}</math> is the moment of the waterplane are (see Chapter 7, Mei (1983)) and <math>I^{V}_{3}</math> is <math>z</math>-component centre of buoyancy of the structure. Thus, the coupled equations of motion for a floating structure subjected to some constraint forces have been derived. (N.B. it is assumed that the centre of rotation and the centre of mass of the structure coincide for this equation, i.e. it is assumed that the body is semi-submerged. Furthermore, any wave incidence is assumed to be along the <math>x</math>-axis.) The scattering and radiation problems are simpler than the coupled problem because the motion of the the structure is then prescribed.
The initial conditions are
{{initial free surface time domain}}
and
{{initial conditions floating body}}

==Frequency domain definitions==

Linearised water-wave problems were first investigated in the frequency domain wherein all motions are assumed to be time-harmonic and to have begun from <math>t= -\infty</math> . The time-dependence can be removed in a straightforward manner by substituting for the potential and any structural motions using
<center><math>
\Phi(\mathbf{x},t)=Re \{\phi(\mathbf{x},\omega) e^{-i\omega t}\}
</math></center>
<center><math>
V(\mathbf{x},t)=Re \{ v(\mathbf{x},\omega) e^{-i\omega t}\}
</math></center>
and this results in a considerable simplification of the problem. The complex amplitudes of the dynamic quantities vary with frequency and by removing the time-dependence the governing equations and boundary conditions will feature these terms only.
However, \citeasnoun{mciver2003} have shown that the frequency-domain potential can also be defined using the following operations on the time-domain solution:

*take a Laplace transform of the time-domain potential
<center><math>
\hat{\phi}(\mathbf{x},s)=\int^{\infty}_{0}\Phi(\mathbf{x},t)e^{-s t}\, dt ,\quad \textrm{Re } s >0;
</math></center>

*apply the change of variables <math>s=-i\omega</math> so that
<center><math>
\phi(\mathbf{x},\omega)=\hat{\phi}(\mathbf{x},-i\omega).
</math></center>

*noting that <math>\phi(\mathbf{x},-\omega)=\bar{\phi}(\mathbf{x},\omega)</math>, the inverse Fourier transform is given by
<center><math>(3)
\Phi(\mathbf{x},t)=\frac{1}{2\pi}\int^{\infty}_{-\infty}\phi(\mathbf{x},\omega)e^{-i\omega t} d\omega =\frac{1}{\pi}Re\int^{\infty}_{0}\phi(\mathbf{x},\omega)e^{-i\omega t}\,d\omega
</math></center>
where the path of integration must pass over any singularities of <math>\phi</math> that lie on the real axis and it has been assumed that there is no motion prior to <math>t=0</math>, i.e. <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math>.

If <math>\omega</math> is real then <math>\phi(\mathbf{x},\omega)</math> is just the frequency domain potential. By the principle of causality, which precludes the existence of the response before the cause so that <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math> , no singularities can exist in <math>\textrm{Im } \omega>0</math>. Therefore, if it is assumed that no singularities exist on the real axis the inverse transform will integrate the frequency-domain potential over the full range of <math>\omega</math>. With this approach, the frequency-domain equations are obtained by applying the Fourier transform to the time-dependent equations. However, unlike the equations resulting from the assumptions~(2) of time-harmonic motion from <math>t= -\infty</math>, the initial conditions will be present in the frequency-domain equations arising from the Fourier transformation. Although, the latter equations are more mathematically consistent it is usual to set all initial conditions to zero before attempting to solve the equations even if this is not the case. Nevertheless, in the frequency-domain analysis of resonances in coupled motion problems \citeasnoun{pmciver2005} retains the initial condition terms in the equation of motion for the structure and obtains some important results regarding the resonant behaviour of the motion. Therefore, the retention of the initial condition terms depends on the application of the subsequent equations.

Many analytical and approximate solutions exist for frequency-domain problems involving structures with simple geometries. Thus, in the simplest frequency-domain radiation and diffraction problems it is possible to completely determine the velocity potential analytically. The hydrodynamic forces, which are crucial in the context of marine engineering, can then be calculated directly from the potential. For more general geometries this is not possible and the hydrodynamic forces must be computed using numerical methods. Nevertheless, it is useful to consider the analytical expressions for the various potentials and forces in the frequency-domain problem.

In a linearised coupled motion problem, it is customary to decompose the total velocity potential into a scattering potential <math>\phi^{S}</math> and a radiation potential <math>\phi^{R}</math>. Thus, the coupled motion problem will require the solution of the radiation problem and the scattering problem. The radiation potential can be further decomposed into a sum over the six modes of motion so that the total frequency domain potential is
<center><math>
\phi=\phi^{S}+\sum_{\mu}v_{\mu}\phi_{\mu}
</math></center>

where <math>u_{\mu}</math> is the complex amplitude of the generalised velocity in the <math>\mu</math> direction and <math>\phi_{\mu}</math> describes the fluid response due to the forced oscillations in mode <math>\mu</math> with unit velocity amplitude. The velocities in each mode will be determined from the frequency domain equation of motion for the body. However, this requires the computation of the radiation and exciting forces on the structure. Therefore, it is necessary to consider the radiation problem prior to the coupled motion equation. It is first important to note that, for a given mode <math>\mu</math> the boundary condition on the structure will be

<center><math>

\frac{\partial\phi_{\mu}}{\partial n}=n_{\mu},

</math></center>

because the total velocity is

<center><math>

v(\omega)=\sum_{\mu}v_{\mu}n_{\mu}

</math></center>

where <math>n_{\mu}</math> is the <math>\mu</math> component of the generalised normal and <math>v_{\mu}</math> is the component of the generalised complex velocity amplitude in this direction. The hydrodynamic force on the structure in the <math>\mu</math> direction due to the fluid response to the forced oscillations is

<center><math>(5)

F_{\mu}^{R}=i\omega\rho\iint_{S_{B}}\phi^{R}n_{\mu} \,dS

</math></center>

and the force due to the diffraction of an incident wave by the fixed structure is

<center><math>(6)

F_{\mu}^{S}=i\omega\rho\iint_{S_{B}}\phi^{S}n_{\mu} \,dS

</math></center>

where the time-dependence has been removed in both cases. By expressing the radiation potential as a sum over the individual modes, the radiation force can written as <math>\sum_{\nu} v_{\nu}f_{\nu\mu}</math> where

<center><math>

f_{\nu\mu}=i\omega\rho\iint_{\Gamma} \phi_{\nu}n_{\mu} \,dS.

</math></center>

It is conventional to decompose the radiation force into a term featuring the added mass matrix <math>a_{\alpha\beta}</math> and a term featuring the damping matrix <math>b_{\alpha\beta}</math> as follows

<center><math>(7)

f_{\beta\alpha}=i\omega(a_{\beta\alpha}+\frac{ib_{\beta\alpha}}{\omega})

</math></center>

where the added mass and damping coefficients are in phase with the acceleration and velocity respectively. These real quantities depend only on the geometry of the structure and the oscillation frequency <math>\omega</math> and describe many of the properties of the structure. In particular, (as will be shown see later) if the variation of these coefficients with frequency is known then we can predict the resonant behaviour of the structure in the time-domain as well as the frequency domain. The computation of these quantities was considered to be very important in marine engineering and naval architecture and so a variety of different computer codes were developed to compute the coefficients efficiently. The boundary element method forms the basis of most of the computer code algorithms, the most widely-used example being WAMIT (see http:<math>\backslash\backslash</math>www.wamit.com) which determines frequency domain solutions for prescribed structures and structure motions.

==Frequency-domain equation of motion==

The radiation force coefficients also play a significant role in determining the motion of a floating structure possibly subject to mooring and applied forces. To obtain the correct equation of motion in the frequency domain it is important to adopt the approach similar to that taken by [[McIver 2005]], i.e. keep the initial condition terms in the Fourier transform operations. Therefore, if the Fourier transform of the velocity is <math>v_{\mu}(\omega)</math> then the Fourier transform of the acceleration is given by

<center><math>

\int_{0}^{\infty}\ddot{X}_{\mu}(t)e^{i\omega t}dt=-i\omega v_{\mu}(\omega) - V_{\mu}(0)

</math></center>

the time-derivative of the potential obeys a similar relation

<center><math>

\int_{0}^{\infty}\frac{\partial\Phi}{\partial t}e^{i\omega t}dt=-i\omega \phi_{\mu}(\omega) - \Phi(\mathbf{x},0).

</math></center>

Therefore, the ini\partial conditions of the structure <math>(X_{\mu}(0),V_{\mu}(0))</math> and of the potential <math>\Phi(\mathbf{x},0)</math> will be present in the Fourier transform of the equation motion which will be referred to as the frequency-domain equation of motion. If the frequency domain potential is decomposed in the same manner as before then the Fourier transform of the equation of motion for the structure is
<center><math>
M\left[-V_{\mu}(0)-i\omega v_{\mu}(\omega)\right]
=-\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS+F^{S}_{\mu}(\omega)+
i\omega \sum_{\nu} (f_{\mu\nu}(\omega)+i\gamma_{\mu\nu}/\omega)v_{\nu}(\omega)-
\sum_{\nu} c_{\mu\nu}\frac{v(\omega)+X(0)}{-i\omega}+f^{A}_{\mu}(\omega)
</math></center>
for <math>\mu=1,\ldots,6</math>, where <math>f^{A}_{\mu}(\omega)</math> is the Fourier transform of the applied force <math>F_{\mu}(t)</math> in equation~(\ref{linearisedmotion}). Although it is assumed that <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math>, for a non-zero initial velocity <math>\lim_{t^{+}\rightarrow 0}\Phi(\mathbf{x},t)\neq 0</math> because of the impulsive pressure on the free-surface resulting from the initial motion of the pressure. Therefore, the first term on the right-hand side of the above equation is not identically zero, rather is given by
<center><math>
\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS=a(\infty)V(0)
</math></center>
[[Mei 1983]]
where <math>a(\infty)=\lim_{\omega\rightarrow\infty}a(\omega)</math> is the infinite frequency added mass.

The frequency-domain equation is usually re-expressed in the following form
<center><math>
\sum_{\nu} \{c_{\mu\nu}-\omega^{2}\left[M_{\mu\mu}\delta_{\mu\nu}+a_{\mu\nu}(\omega)+i(b_{\mu\nu}+\gamma_{\mu\nu})/\omega\right]\}v_{\nu}(\omega)
= -i\omega\left[ F_{\mu}^{S}(\omega)+f_{\mu}(\omega)+(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)\right]-\sum_{\nu}c_{\mu\nu}X_{\nu}(0)
</math></center>
for <math>\mu=1,\ldots,6</math>. If it is assumed that the structure can only move in one mode of motion and that there is no incident wave or applied force then the complex velocity will be given by
<center><math>
v_{\mu}(\omega)=\frac{-i\omega(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)-c_{\mu\mu}X_{\mu}(0)}{c_{\mu\mu}-\omega^{2}\left[M_{\mu\mu}+a_{\mu\mu}(\omega)+i(b_{\mu\mu}+\gamma_{\mu\mu})/\omega\right]}.
</math></center>

As described by [[McIver 2006]], the zero of the denominator gives the location of the pole referred to as the motion resonance. The complex force coefficient <math>f_{\mu\nu}(\omega)</math> will also possess a pole, however this resonance is annulled by the frequency-dependence of the complex velocity (see [[McIver 2006]]) and so the motion, in the absence of incident waves, is dominated by the motion resonance term. Solving for <math>v(\omega)</math> then the time-domain velocity can be recovered using the inverse transform
<center><math>
V(t)=\frac{1}{\pi} Re \int^{\infty}_{0}v(\omega)e^{-i\omega t} dt.
</math></center>

==Integro-differential equation method==

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by <math>\delta(t-\tau)V_{\alpha}(\tau)</math> where <math>\dot{X_{\alpha}}=V_{\alpha}</math>. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function <math>V_{\alpha}(t)</math> an integral of the impulse response over the range <math>-\infty<\tau<\infty</math> must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

<center><math>(11)

\mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau

</math></center>

where <math>m_{\beta\alpha}(\infty)</math> is an infinite frequency added mass coefficient and <math>L_{\beta\alpha}(t-\tau)</math> is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. <math>V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \}</math>, the impulse response function <math>L_{\beta\alpha}</math> can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

<center><math>

\begin{split}

a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, d\tau \

b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, d\tau

\end{split}

</math></center>

and so if the added mass or damping coefficients are known for all frequencies <math>0\leq\omega\leq\infty</math>, the impulse response or memory function <math>L_{\alpha\beta}(t)</math> can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

<center><math>

\left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t),

</math></center>
for <math>\alpha=1,\ldots,6</math>, where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency <math>\omega</math> on the interval <math>[0,\infty)</math>. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

==Generalised eigenfunction expansion method==

The singularity expansion method (SEM) is based on a general scattering theory for general wave problems and has been adapted by \citeasnoun{hazardSEM} and \citeasnoun{meylaniwwwfb2008}

to problems in hydrodynamics and hydro-elasticity. The central concept in the theory is to express the governing equations for the scattering problem in terms of an evolution equation involving a unitary operator and to use the generalised eigenfunctions and eigenfrequencies of this operator to give an approximation to the time-domain solution of the scattering problem. To give a brief overview of the method the scattering equations must be re-expressed using a harmonic lifting operator <math>\mathbf{B}</math> which maps a function defined on the free-surface onto the total fluid domain. This harmonic lifting operator is denoted by <math>\mathbf{B}</math> so that <math>\mathbf{B}\Psi=\Phi</math> where <math>\Psi=\Phi</math> on the free-surface <math>F</math>. If <math>D</math> denotes the total fluid domain, then potential <math>\Phi</math> satisfies

<center><math>\begin{matrix}

\nabla^{2}\Phi&=0 \quad\textrm{in } D, \

\frac{\partial\Phi}{\partial n} &= 0 \quad\textrm{on } \partial D,\

\Phi&=\Psi \quad\textrm{on } F

\end{matrix}</math></center>

where <math>\partial D</math> includes the bed and the structure surface. The Dirichlet to Neumann map <math>\partial_{\mathbf{n}}\mathbf{B}</math> defined by

<center><math>(12)

\partial_{\mathbf{n}}\mathbf{B}\Psi=\frac{\partial\Phi}{\partial z}, \quad x\in F,

</math></center>

then recovers the corresponding normal derivative to <math>\Psi</math> on the free surface.

The evolution equation

<center><math>

i\frac{\partial \mathbf{U}}{\partial t} = \mathcal{A}\mathbf{U}

</math></center>

accounts for the linear time-dependent free-surface equations using the vector

<center><math>

\mathbf{U}= \begin{pmatrix}

\Psi \

-i\eta

\end{pmatrix},

</math></center>

with <math>\eta</math> denoting the time-dependent free-surface elevation, and the operator <math>\mathcal{A}</math> which is both unitary and self-adjoint given by

<center><math>

\mathcal{A}= \begin{pmatrix}

0 & g \

\partial_{\mathbf{n}}\mathbf{B} & 0 \

\end{pmatrix}

</math></center>

A self-adjoint operator will possess an entirely real spectrum and the generalised eigenfunctions <math>\mathbf{u}=(\psi,-i\zeta)^{T}</math>, which are non-trivial solutions of

<center><math>(13)

\mathcal{A}\mathbf{u}=\omega\mathbf{u},

</math></center>

are just frequency-domain scattering solutions. (On a technical note, the word `generalised' precedes eigenfunction because the energy is unbounded.) This can be shown easily by combining the scalar equations resulting from (13) so as to give the frequency-domain free-surface condition. For a given frequency <math>\omega</math>, the free-surface condition in three-dimensions is satisfied by waves from an infinite number of directions and so for each <math>\omega</math> there is an infinite set of eigenfunctions <math>\mathbf{u}_{n}=(1,\omega/g)^{T}\psi_{n}(\mathbf{x},\omega)</math>, with the <math>n^{th}</math> eigenfunction corresponding to incident waves of the form <math>J_{n}(k r) e^{in\theta}</math>. The general solution of the time-evolution equation is, from spectral theory,

<center><math>(14)

\mathbf{U}(\mathbf{x},t)=\int^{\infty}_{-\infty}\left[ \sum_{n}f_{n}(\omega)\mathbf{u}_{n}(\mathbf{x},\omega) \right]e^{-i\omega t}\, d\omega

</math></center>

where <math>f_{n}(\omega)</math> is determined by the initial conditions now expressed as <math>\mathbf{U}(\mathbf{x},0)</math>. To obtain this expression for <math>f_{n}(\omega)</math>, apply the energy inner product to (14) evaluated at <math>t=0</math>. The eigenfunctions satisfy the orthogonality condition

<center><math>(15)

<\mathbf{u}_{m}(\mathbf{x},\omega),\mathbf{u}_{n}(\mathbf{x},\omega')>_{E}=\Lambda_{m}(\omega)\delta_{mn}\delta(\omega-\omega')

</math></center>

where <math><>_{E}</math> denotes a special energy inner product (defined with a <math>\mathcal{H}</math> subscript by \citeasnoun{meylaniwwwfb2008}) and it can be shown that

<center><math>

\Lambda_{m}(\omega)=\frac{4\pi\omega^{2}}{g k}\frac{d\omega}{dk}.

</math></center>

Therefore, it is straightforward to show that

<center><math>

f_{n}(\omega)=\frac{1}{\Lambda_{n}(\omega)}<\mathbf{U}(\mathbf{x},0),\mathbf{u}_{n}(\mathbf{x},\omega)>_{E}.

</math></center>

by evaluating the energy inner product <math><\mathbf{U}(\mathbf{x},0),\mathbf{u}_{n}(\mathbf{x},\omega)>_{E}</math> using the orthogonality relation~(15) and the definition~(14). This expression can be further simplified using the definition of the energy product but details will not be provided here. Instead, it should be noted that an analytic expression for the general solution has been obtained. Thus, for a given scattering problem it is in theory possible to obtain the motion of the free-surface and the potential on the free-surface.

The singularity expansion method involves moving the path of integration in~(14) in the <math>\omega</math>-plane across the singularities in the lower half plane Im<math>\,\omega\leq0</math> using the method of contour integration. Thus, the general solution will consist of contributions from the poles, any branch cuts, the path at infinity and the remainder of the path located below the poles in the <math>\omega</math>-plane. As explained by \citeasnoun{hazardSEM}, the contributions from infinity are assumed to be identically zero and the contribution from the remainder of the path decays faster than the pole contributions. Given that the contribution from the branch cuts are thought to be significant only for very large times an approximation to the general solution can be obtained for medium and large <math>t</math> because the contributions from the poles dominate for this range of times. Therefore, the SEM requires a knowledge of the pole structure of the integrand to approximate the complete integral. Furthermore, the behaviour of the eigenfunctions in the vicinity of the poles must be known in order to compute the residue of each pole. The scattering potential will satisfy

<center><math>

(I+T(\omega))\phi^{S}=g_{I}

</math></center>

where the form of the operator <math>T(\omega)</math> depends on the solution method chosen and <math>g_{I}</math> is determined by the incident wave. Given <math>g_{I}</math>, the scattered field is

<center><math>

\phi^{S}=(I+T(\omega))^{-1}g_{I}

</math></center>

and the pole structure of <math>\phi_{S}</math> will be inherited from the operator <math>(I+T(\omega))^{-1}</math>, referred to as the resolvent.

In the case where the resolvent operator is approximated by a matrix, it can be shown that the potential has the form

<center><math>(16)

\phi_{n}(\mathbf{x},\omega)\sim\frac{\alpha_{nj}v_{j}(\mathbf{x})}{\omega-\omega_{j}} \textrm{ as } \omega\rightarrow\omega_{j}

</math></center>

in the vicinity of the pole. The poles themselves are determined by locating values of <math>\omega</math> in the lower complex plane where the resolvent is not invertible. Thus, <math>v_{j}</math> is a generalised eigenfunction of <math>A_{j0}=I+T(\omega_{j})</math>. The integral in equation~(14) can be approximated by closing the integration path in the lower half plane and then moving the integration path across a finite number of <math>P</math> poles and summing over the contributions of these poles. As described by~\citeasnoun{hazardSEM}, any branch cuts in the complex <math>\omega</math>-plane will only be significant at very large times and the integrals at infinity are expected to be zero. Therefore, the contribution of the <math>P</math> poles closest to the real-<math>\omega</math> axis (and hence with the smallest decay rates <math>e^{-Im(\omega_{j})}</math>) will dominate in the medium term. So, after neglecting other contributions to the integral and using~(16) to determine the residues of the poles, the potential on the free-surface will be given by

<center><math>(17)

\Psi(\mathbf{x},t)\approx -2\pi\sum_{j}\left[ \sum_{n}f_{n}(\omega_{j})\alpha_{nj}\right]v_{j}(\mathbf{x},0)e^{-i\omega_{j}t}.

</math></center>

Although full details of how this expression can be evaluated are not given here, examples of applications of this method are given by~\citeasnoun{meylaniwwwfb2007}, \citeasnoun{meylaniwwwfb2008} and \citeasnoun{meylan2002}. Each of these papers also contain more detailed instructions regarding the computational aspects of the method, such as determining the locations of the poles.

\citeasnoun{hazardSEM} notes that the SEM is a non-rigorous method and requires some heuristic arguments to justify its use. However, in the specific cases considered the results are generally accurate for a large range of times and much frequency-domain information regarding resonances is inherent in the time-domain solution. At present, the method only applies to scattering problems; however, it is hoped that it can be extended to radiation and coupled motion problems. It is not clear how the forces on the structure will be determined.

\f1

}

{\rtf1\ansi\ansicpg1252\deff0\deflang2057{\fonttbl{\f0\fnil\fcharset0 Courier New;}{\f1\fswiss\fcharset0 Arial;}}

{\*\generator Msftedit 5.41.21.2508;}\viewkind4\uc1d\f0\fs20=Review of time-domain models=

==Frequency domain definitions==(1)

Linearised water-wave problems were first investigated in the frequency domain wherein all motions are assumed to be time-harmonic and to have begun from <math>t= -\infty</math> . The time-dependence can be removed in a straightforward manner by substituting for the potential and any structural motions using

<center><math>(2)

\begin{split}

\Phi(\mbf{x},t)=Re \{\phi(\mbf{x},\omega) e^{-i\omega t}\} \

V(\mbf{x},t)=Re \{ v(\mbf{x},\omega) e^{-i\omega t}\}

\end{split}

</math></center>

and this results in a considerable simplification of the problem. The complex amplitudes of the dynamic quantities vary with frequency and by removing the time-dependence the governing equations and boundary conditions will feature these terms only.
However, \citeasnoun{mciver2003} have shown that the frequency-domain potential can also be defined using the following operations on the time-domain solution:

*take a Laplace transform of the time-domain potential
<center><math>
\hat{\phi}(\mbf{x},s)=\int^{\infty}_{0}\Phi(\mbf{x},t)e^{-s t}\, dt ,\quad \textrm{Re } s >0;
</math></center>

*apply the change of variables <math>s=-i\omega</math> so that
<center><math>
\phi(\mbf{x},\omega)=\hat{\phi}(\mbf{x},-i\omega).
</math></center>

*noting that <math>\phi(\mbf{x},-\omega)=\bar{\phi}(\mbf{x},\omega)</math>, the inverse Fourier transform is given by
<center><math>(3)
\Phi(\mbf{x},t)=\frac{1}{2\pi}\int^{\infty}_{-\infty}\phi(\mbf{x},\omega)e^{-i\omega t} d\omega =\frac{1}{\pi}Re\int^{\infty}_{0}\phi(\mbf{x},\omega)e^{-i\omega t}\,d\omega
</math></center>
where the path of integration must pass over any singularities of <math>\phi</math> that lie on the real axis and it has been assumed that there is no motion prior to <math>t=0</math>, i.e. <math>\Phi(\mbf{x},t)=0</math> for <math>t<0</math>.

If <math>\omega</math> is real then <math>\phi(\mbf{x},\omega)</math> is just the frequency domain potential. By the principle of causality, which precludes the existence of the response before the cause so that <math>\Phi(\mbf{x},t)=0</math> for <math>t<0</math> , no singularities can exist in <math>\textrm{Im } \omega>0</math>. Therefore, if it is assumed that no singularities exist on the real axis the inverse transform will integrate the frequency-domain potential over the full range of <math>\omega</math>. With this approach, the frequency-domain equations are obtained by applying the Fourier transform to the time-dependent equations. However, unlike the equations resulting from the assumptions~(2) of time-harmonic motion from <math>t= -\infty</math>, the initial conditions will be present in the frequency-domain equations arising from the Fourier transformation. Although, the latter equations are more mathematically consistent it is usual to set all initial conditions to zero before attempting to solve the equations even if this is not the case. Nevertheless, in the frequency-domain analysis of resonances in coupled motion problems \citeasnoun{pmciver2005} retains the initial condition terms in the equation of motion for the structure and obtains some important results regarding the resonant behaviour of the motion. Therefore, the retention of the initial condition terms depends on the application of the subsequent equations.

Many analytical and approximate solutions exist for frequency-domain problems involving structures with simple geometries. Thus, in the simplest frequency-domain radiation and diffraction problems it is possible to completely determine the velocity potential analytically. The hydrodynamic forces, which are crucial in the context of marine engineering, can then be calculated directly from the potential. For more general geometries this is not possible and the hydrodynamic forces must be computed using numerical methods. Nevertheless, it is useful to consider the analytical expressions for the various potentials and forces in the frequency-domain problem.

In a linearised coupled motion problem, it is customary to decompose the total velocity potential into a scattering potential <math>\phi^{S}</math> and a radiation potential <math>\phi^{R}</math>. Thus, the coupled motion problem will require the solution of the radiation problem and the scattering problem. The radiation potential can be further decomposed into a sum over the six modes of motion so that the total frequency domain potential is

<center><math>(4)

\phi=\phi^{S}+\sum_{\mu}v_{\mu}\phi_{\mu}

</math></center>

where <math>u_{\mu}</math> is the complex amplitude of the generalised velocity in the <math>\mu</math> direction and <math>\phi_{\mu}</math> describes the fluid response due to the forced oscillations in mode <math>\mu</math> with unit velocity amplitude. The velocities in each mode will be determined from the frequency domain equation of motion for the body. However, this requires the computation of the radiation and exciting forces on the structure. Therefore, it is necessary to consider the radiation problem prior to the coupled motion equation. It is first important to note that, for a given mode <math>\mu</math> the boundary condition on the structure will be

<center><math>

\frac{tial\phi_{\mu}}{tial n}=n_{\mu},

</math></center>

because the total velocity is

<center><math>

v(\omega)=\sum_{\mu}v_{\mu}n_{\mu}

</math></center>

where <math>n_{\mu}</math> is the <math>\mu</math> component of the generalised normal and <math>v_{\mu}</math> is the component of the generalised complex velocity amplitude in this direction. The hydrodynamic force on the structure in the <math>\mu</math> direction due to the fluid response to the forced oscillations is

<center><math>(5)

F_{\mu}^{R}=i\omega\rho\iint_{S_{B}}\phi^{R}n_{\mu} \,dS

</math></center>

and the force due to the diffraction of an incident wave by the fixed structure is

<center><math>(6)

F_{\mu}^{S}=i\omega\rho\iint_{S_{B}}\phi^{S}n_{\mu} \,dS

</math></center>

where the time-dependence has been removed in both cases. By expressing the radiation potential as a sum over the individual modes, the radiation force can written as <math>\sum_{\nu} v_{\nu}f_{\nu\mu}</math> where

<center><math>

f_{\nu\mu}=i\omega\rho\iint_{\Gamma} \phi_{\nu}n_{\mu} \,dS.

</math></center>

It is conventional to decompose the radiation force into a term featuring the added mass matrix <math>a_{\alpha\beta}</math> and a term featuring the damping matrix <math>b_{\alpha\beta}</math> as follows

<center><math>(7)

f_{\beta\alpha}=i\omega(a_{\beta\alpha}+\frac{ib_{\beta\alpha}}{\omega})

</math></center>

where the added mass and damping coefficients are in phase with the acceleration and velocity respectively. These real quantities depend only on the geometry of the structure and the oscillation frequency <math>\omega</math> and describe many of the properties of the structure. In particular, (as will be shown see later) if the variation of these coefficients with frequency is known then we can predict the resonant behaviour of the structure in the time-domain as well as the frequency domain. The computation of these quantities was considered to be very important in marine engineering and naval architecture and so a variety of different computer codes were developed to compute the coefficients efficiently. The boundary element method forms the basis of most of the computer code algorithms, the most widely-used example being WAMIT (see http:<math>\backslash\backslash</math>www.wamit.com) which determines frequency domain solutions for prescribed structures and structure motions.

==Frequency-domain equation of motion==(8)

The radiation force coefficients also play a significant role in determining the motion of a floating structure possibly subject to mooring and applied forces. To obtain the correct equation of motion in the frequency domain it is important to adopt the approach similar to that taken by~\citeasnoun{pmciver2005}, i.e. keep the initial condition terms in the Fourier transform operations. Therefore, if the Fourier transform of the velocity is <math>v_{\mu}(\omega)</math> then the Fourier transform of the acceleration is given by

<center><math>

\int_{0}^{\infty}\ddot{X}_{\mu}(t)e^{i\omega t}dt=-i\omega v_{\mu}(\omega) - V_{\mu}(0)

</math></center>

the time-derivative of the potential obeys a similar relation

<center><math>

\int_{0}^{\infty}\frac{tial\Phi}{tial t}e^{i\omega t}dt=-i\omega \phi_{\mu}(\omega) - \Phi(\mbf{x},0).

</math></center>

Therefore, the initial conditions of the structure <math>(X_{\mu}(0),V_{\mu}(0))</math> and of the potential <math>\Phi(\mbf{x},0)</math> will be present in the Fourier transform of the equation motion which will be referred to as the frequency-domain equation of motion. If the frequency domain potential is decomposed in the same manner as before~(4) then the Fourier transform of the equation of motion~(\ref{linearisedmotion}) for the structure is

<center><math>(9)

\begin{split}

M\left[-V_{\mu}(0)-i\omega v_{\mu}(\omega)\right]&

=-\rho\iint_{\Gamma}(-\Phi(\mbf{x},0))n_{\mu}dS+F^{S}_{\mu}(\omega)+\

i\omega \sum_{\nu}&(f_{\mu\nu}(\omega)+i\gamma_{\mu\nu}/\omega)v_{\nu}(\omega)-

\sum_{\nu} c_{\mu\nu}\frac{v(\omega)+X(0)}{-i\omega}+f^{A}_{\mu}(\omega)

\end{split}

</math></center>

for <math>\mu=1,\ldots,6</math>, where <math>f^{A}_{\mu}(\omega)</math> is the Fourier transform of the applied force <math>F_{\mu}(t)</math> in equation~(\ref{linearisedmotion}). Although it is assumed that <math>\Phi(\mbf{x},t)=0</math> for <math>t<0</math>, for a non-zero initial velocity <math>\lim_{t^{+}\rightarrow 0}\Phi(\mbf{x},t)\neq 0</math> because of the impulsive pressure on the free-surface resulting from the initial motion of the pressure. Therefore, the first term on the right-hand side of the above equation is not identically zero, rather is given by

<center><math>

\rho\iint_{\Gamma}(-\Phi(\mbf{x},0))n_{\mu}dS=a(\infty)V(0)

</math></center>

(see \citeasnoun{ccmei2}, \S~8.12.1)

where <math>a(\infty)=\lim_{\omega\rightarrow\infty}a(\omega)</math> is the infinite frequency added mass.

The frequency-domain equation is usually re-expressed in the following form

<center><math>

\begin{split}

\sum_{\nu}&\{c_{\mu\nu}-\omega^{2}\left[M_{\mu\mu}\delta_{\mu\nu}+a_{\mu\nu}(\omega)+i(b_{\mu\nu}+\gamma_{\mu\nu})/\omega\right]\}v_{\nu}(\omega) \

=& -i\omega\left[ F_{\mu}^{S}(\omega)+f_{\mu}(\omega)+(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)\right]-\sum_{\nu}c_{\mu\nu}X_{\nu}(0)\end{split}

</math></center>

for <math>\mu=1,\ldots,6</math>. If it is assumed that the structure can only move in one mode of motion and that there is no incident wave or applied force then the complex velocity will be given by

<center><math>(10)

v_{\mu}(\omega)=\frac{-i\omega(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)-c_{\mu\mu}X_{\mu}(0)}{c_{\mu\mu}-\omega^{2}\left[M_{\mu\mu}+a_{\mu\mu}(\omega)+i(b_{\mu\mu}+\gamma_{\mu\mu})/\omega\right]}.

</math></center>

As described by \citeasnoun{mciver2006}, the zero of the denominator gives the location of the pole referred to as the motion resonance. The complex force coefficient <math>f_{\mu\nu}(\omega)</math> will also possess a pole, however this resonance is annulled by the frequency-dependence of the complex velocity (see \S~6 of \citeasnoun{mciver2006}) and so the motion, in the absence of incident waves, is dominated by the motion resonance term. Solving equation~(10) for <math>v(\omega)</math> then the time-domain velocity can be recovered using the inverse transform

<center><math>

V(t)=\frac{1}{\pi} Re \int^{\infty}_{0}v(\omega)e^{-i\omega t} dt.

</math></center>

==Integro-differential equation method==

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by <math>\delta(t-\tau)V_{\alpha}(\tau)</math> where <math>\dot{X_{\alpha}}=V_{\alpha}</math>. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function <math>V_{\alpha}(t)</math> an integral of the impulse response over the range <math>-\infty<\tau<\infty</math> must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

<center><math>(11)

\mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau

</math></center>

where <math>m_{\beta\alpha}(\infty)</math> is an infinite frequency added mass coefficient and <math>L_{\beta\alpha}(t-\tau)</math> is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. <math>V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \}</math>, the impulse response function <math>L_{\beta\alpha}</math> can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

<center><math>

\begin{split}

a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, d\tau \

b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, d\tau

\end{split}

</math></center>

and so if the added mass or damping coefficients are known for all frequencies <math>0\leq\omega\leq\infty</math>, the impulse response or memory function <math>L_{\alpha\beta}(t)</math> can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

<center><math>

\left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t),

</math></center>
for <math>\alpha=1,\ldots,6</math>, where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency <math>\omega</math> on the interval <math>[0,\infty)</math>. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

==Generalised eigenfunction expansion method==

The singularity expansion method (SEM) is based on a general scattering theory for general wave problems and has been adapted by \citeasnoun{hazardSEM} and \citeasnoun{meylaniwwwfb2008}

to problems in hydrodynamics and hydro-elasticity. The central concept in the theory is to express the governing equations for the scattering problem in terms of an evolution equation involving a unitary operator and to use the generalised eigenfunctions and eigenfrequencies of this operator to give an approximation to the time-domain solution of the scattering problem. To give a brief overview of the method the scattering equations must be re-expressed using a harmonic lifting operator <math>\mbf{B}</math> which maps a function defined on the free-surface onto the total fluid domain. This harmonic lifting operator is denoted by <math>\mbf{B}</math> so that <math>\mbf{B}\Psi=\Phi</math> where <math>\Psi=\Phi</math> on the free-surface <math>F</math>. If <math>D</math> denotes the total fluid domain, then potential <math>\Phi</math> satisfies

<center><math>\begin{matrix}

\nabla^{2}\Phi&=0 \quad\textrm{in } D, \

\frac{tial\Phi}{tial n} &= 0 \quad\textrm{on } tial D,\

\Phi&=\Psi \quad\textrm{on } F

\end{matrix}</math></center>

where <math>tial D</math> includes the bed and the structure surface. The Dirichlet to Neumann map <math>tial_{\mbf{n}}\mbf{B}</math> defined by

<center><math>(12)

tial_{\mbf{n}}\mbf{B}\Psi=\frac{tial\Phi}{tial z}, \quad x\in F,

</math></center>

then recovers the corresponding normal derivative to <math>\Psi</math> on the free surface.

The evolution equation

<center><math>

i\frac{tial \mbf{U}}{tial t} = \mathcal{A}\mbf{U}

</math></center>

accounts for the linear time-dependent free-surface equations using the vector

<center><math>

\mbf{U}= \begin{pmatrix}

\tab \Psi \

\tab -i\eta

\end{pmatrix},

</math></center>

with <math>\eta</math> denoting the time-dependent free-surface elevation, and the operator <math>\mathcal{A}</math> which is both unitary and self-adjoint given by

<center><math>

[[Category:Linear Water-Wave Theory]]

{{complete page}}

Category:Time-Dependent Linear Water Waves

2009-08-20T23:09:51Z

Cfitzgerald: /* Time domain problem for a floating structure */

Generally the focus of research is on the [[Frequency Domain Problem]]. The time-domain
problem can be solved by [[Generalised Eigenfunction Expansion]] or by
an [[Expansion over the Resonances]] or using [[Memory Effect Function]].

== Introduction ==

Analytical solutions of the linearised time-domain equations for a coupled-motion or forced-motion problems are very rare. In fact, only a small number of analytic time-domain solutions have been obtained (see the papers by [[Kennard 1949]] and [[McIver 1994]]) for structures with simple geometries. Therefore, a variety of numerical methods have been developed to solve initial-value time-domain wave-structure interaction problems. The Fourier transform enables the time and frequency domain quantities to be related and the frequency domain forces and potentials have important roles in the development of the numerical time-domain solution methods. This can be partly attributed to the fact that, in the past, analytical or semi-analytical solutions to linearised water-wave problems were more easily obtained in the frequency domain given that the assumption of time-harmonic motion results in a significant simplification of the interaction equations. Therefore, it was natural to develop time-domain solutions based on frequency domain results and the Fourier transform relation between the domains.

== Time domain problem for a floating structure ==
To begin with, it is useful to introduce the general time-dependent water wave problem. Therefore, the the time-domain equations for the motion of a floating structure must be quoted. From these equations, the forced motion and fixed body interaction problems (i.e. the radiation and scattering problems) can be derived. Furthermore, the corresponding frequency-domain equations can be derived from the time-domain in order to illustrate the important relation between the two. For a semi-submerged, vertically axisymmetric structure the time-domain linearised equations are
<center><math>
\Delta\Phi\left( \mathbf{x,}t\right) =0,\ \ \mathbf{x}\in\Omega,
</math></center>
<center><math>
\partial_{n}\Phi=0,\ \ z=-h,
</math></center>
<center><math>
\partial_{n}\Phi=\sum\dot{X}_{\nu}n_{\nu},\ \mathbf{x}\in\partial\Omega,
</math></center>
where <math>\Phi</math> is the velocity potential for the fluid, <math>\Gamma</math> is the still water position of the wetted surface of the structure. At the free surface <math>F</math>, the potential must obey the kinematic condition
<center><math>
\partial_{t}\zeta=\partial_{n}\Phi,\ \ z=0,\ x\in F,
</math></center>
and the dynamic condition (the linearized Bernoulli equation)
<center><math>
\zeta = -(1/g)\partial_{t}\Phi,\ \ z=0,\ x\in F,
</math></center>
where <math>\zeta</math> is the free-surface elevation. Initial conditions for <math>\Phi</math> and <math>\partial_{t}\Phi</math> must also be specified. These fluid motion equations must be combined with the equation of motion of the body to give the coupled motion problem involving the structure motions, described by the displacement <math>X_{\nu}</math> in mode <math> \nu</math>, and the velocity potential <math>\Phi</math>. The equation of motion for a structure moored by an arrangement of linear springs and dampers is
<center><math>
M_{\mu\mu}\ddot{X}_{\mu}=-\rho\iint_{\Gamma} \frac{\partial\Phi}{\partial t} n_{\mu}\, dS-\sum^{6}_{\nu=1}\left[ c_{\mu\nu}X_{\nu}(t)+\gamma_{\mu\nu}\dot{X}_{\nu}\right]+F_{\mu}(t), \qquad \mu=1,\ldots,6
</math></center>
where <math>M_{\mu\mu}</math> are the diagonal elements of mass matrix for the structure, i.e. <math>M_{\mu\mu}=M</math> for <math>\mu=1,2,3</math> and <math>M_{\mu\mu}=I_{\mu-3,\mu-3}</math> for <math>\mu=4,5,6</math>.The characteristics of the mooring springs and dampers are described the matrices <math>\kappa_{\mu\nu}</math> and <math>\gamma_{\mu\nu}</math> , with the spring term included in the term <math>c_{\mu\nu}=\rho g b_{\mu}\delta_{\mu\nu}+k_{\mu\nu}</math> which also describes the effect of buoyancy. The initial generalised displacements <math>X_{\mu}</math> and velocities <math>\dot{X}_{\mu}</math> of the body must be specified for all modes <math>\mu=1,\ldots,6</math> in order to solve the equation.
It is non-zero for the heave, roll and pitch modes only and these terms are <math>b_{3}=W</math>, <math>b_{4}=I^{A}_{22}+I^{V}_{3}</math> and <math>b_{5}=I^{A}_{11}+I^{V}_{3}</math> respectively, where <math>W</math> is the waterplane area and <math>I^{A}</math> is the moment of the waterplane are (see Chapter 7, Mei (1983)) and <math>I^{V}_{3}</math> is <math>z</math>-component centre of buoyancy of the structure. Thus, the coupled equations of motion for a floating structure subjected to some constraint forces have been derived. (N.B. it is assumed that the centre of rotation and the centre of mass of the structure coincide for this equation, i.e. it is assumed that the body is semi-submerged. Furthermore, any wave incidence is assumed to be along the <math>x</math>-axis.) The scattering and radiation problems are simpler than the coupled problem because the motion of the the structure is then prescribed.

{{Equations for fixed bodies in the time domain}}

==Frequency domain definitions==

Linearised water-wave problems were first investigated in the frequency domain wherein all motions are assumed to be time-harmonic and to have begun from <math>t= -\infty</math> . The time-dependence can be removed in a straightforward manner by substituting for the potential and any structural motions using
<center><math>
\Phi(\mathbf{x},t)=Re \{\phi(\mathbf{x},\omega) e^{-i\omega t}\}
</math></center>
<center><math>
V(\mathbf{x},t)=Re \{ v(\mathbf{x},\omega) e^{-i\omega t}\}
</math></center>
and this results in a considerable simplification of the problem. The complex amplitudes of the dynamic quantities vary with frequency and by removing the time-dependence the governing equations and boundary conditions will feature these terms only.
However, \citeasnoun{mciver2003} have shown that the frequency-domain potential can also be defined using the following operations on the time-domain solution:

*take a Laplace transform of the time-domain potential
<center><math>
\hat{\phi}(\mathbf{x},s)=\int^{\infty}_{0}\Phi(\mathbf{x},t)e^{-s t}\, dt ,\quad \textrm{Re } s >0;
</math></center>

*apply the change of variables <math>s=-i\omega</math> so that
<center><math>
\phi(\mathbf{x},\omega)=\hat{\phi}(\mathbf{x},-i\omega).
</math></center>

*noting that <math>\phi(\mathbf{x},-\omega)=\bar{\phi}(\mathbf{x},\omega)</math>, the inverse Fourier transform is given by
<center><math>(3)
\Phi(\mathbf{x},t)=\frac{1}{2\pi}\int^{\infty}_{-\infty}\phi(\mathbf{x},\omega)e^{-i\omega t} d\omega =\frac{1}{\pi}Re\int^{\infty}_{0}\phi(\mathbf{x},\omega)e^{-i\omega t}\,d\omega
</math></center>
where the path of integration must pass over any singularities of <math>\phi</math> that lie on the real axis and it has been assumed that there is no motion prior to <math>t=0</math>, i.e. <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math>.

If <math>\omega</math> is real then <math>\phi(\mathbf{x},\omega)</math> is just the frequency domain potential. By the principle of causality, which precludes the existence of the response before the cause so that <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math> , no singularities can exist in <math>\textrm{Im } \omega>0</math>. Therefore, if it is assumed that no singularities exist on the real axis the inverse transform will integrate the frequency-domain potential over the full range of <math>\omega</math>. With this approach, the frequency-domain equations are obtained by applying the Fourier transform to the time-dependent equations. However, unlike the equations resulting from the assumptions~(2) of time-harmonic motion from <math>t= -\infty</math>, the initial conditions will be present in the frequency-domain equations arising from the Fourier transformation. Although, the latter equations are more mathematically consistent it is usual to set all initial conditions to zero before attempting to solve the equations even if this is not the case. Nevertheless, in the frequency-domain analysis of resonances in coupled motion problems \citeasnoun{pmciver2005} retains the initial condition terms in the equation of motion for the structure and obtains some important results regarding the resonant behaviour of the motion. Therefore, the retention of the initial condition terms depends on the application of the subsequent equations.

Many analytical and approximate solutions exist for frequency-domain problems involving structures with simple geometries. Thus, in the simplest frequency-domain radiation and diffraction problems it is possible to completely determine the velocity potential analytically. The hydrodynamic forces, which are crucial in the context of marine engineering, can then be calculated directly from the potential. For more general geometries this is not possible and the hydrodynamic forces must be computed using numerical methods. Nevertheless, it is useful to consider the analytical expressions for the various potentials and forces in the frequency-domain problem.

In a linearised coupled motion problem, it is customary to decompose the total velocity potential into a scattering potential <math>\phi^{S}</math> and a radiation potential <math>\phi^{R}</math>. Thus, the coupled motion problem will require the solution of the radiation problem and the scattering problem. The radiation potential can be further decomposed into a sum over the six modes of motion so that the total frequency domain potential is
<center><math>
\phi=\phi^{S}+\sum_{\mu}v_{\mu}\phi_{\mu}
</math></center>

where <math>u_{\mu}</math> is the complex amplitude of the generalised velocity in the <math>\mu</math> direction and <math>\phi_{\mu}</math> describes the fluid response due to the forced oscillations in mode <math>\mu</math> with unit velocity amplitude. The velocities in each mode will be determined from the frequency domain equation of motion for the body. However, this requires the computation of the radiation and exciting forces on the structure. Therefore, it is necessary to consider the radiation problem prior to the coupled motion equation. It is first important to note that, for a given mode <math>\mu</math> the boundary condition on the structure will be

<center><math>

\frac{\partial\phi_{\mu}}{\partial n}=n_{\mu},

</math></center>

because the total velocity is

<center><math>

v(\omega)=\sum_{\mu}v_{\mu}n_{\mu}

</math></center>

where <math>n_{\mu}</math> is the <math>\mu</math> component of the generalised normal and <math>v_{\mu}</math> is the component of the generalised complex velocity amplitude in this direction. The hydrodynamic force on the structure in the <math>\mu</math> direction due to the fluid response to the forced oscillations is

<center><math>(5)

F_{\mu}^{R}=i\omega\rho\iint_{S_{B}}\phi^{R}n_{\mu} \,dS

</math></center>

and the force due to the diffraction of an incident wave by the fixed structure is

<center><math>(6)

F_{\mu}^{S}=i\omega\rho\iint_{S_{B}}\phi^{S}n_{\mu} \,dS

</math></center>

where the time-dependence has been removed in both cases. By expressing the radiation potential as a sum over the individual modes, the radiation force can written as <math>\sum_{\nu} v_{\nu}f_{\nu\mu}</math> where

<center><math>

f_{\nu\mu}=i\omega\rho\iint_{\Gamma} \phi_{\nu}n_{\mu} \,dS.

</math></center>

It is conventional to decompose the radiation force into a term featuring the added mass matrix <math>a_{\alpha\beta}</math> and a term featuring the damping matrix <math>b_{\alpha\beta}</math> as follows

<center><math>(7)

f_{\beta\alpha}=i\omega(a_{\beta\alpha}+\frac{ib_{\beta\alpha}}{\omega})

</math></center>

where the added mass and damping coefficients are in phase with the acceleration and velocity respectively. These real quantities depend only on the geometry of the structure and the oscillation frequency <math>\omega</math> and describe many of the properties of the structure. In particular, (as will be shown see later) if the variation of these coefficients with frequency is known then we can predict the resonant behaviour of the structure in the time-domain as well as the frequency domain. The computation of these quantities was considered to be very important in marine engineering and naval architecture and so a variety of different computer codes were developed to compute the coefficients efficiently. The boundary element method forms the basis of most of the computer code algorithms, the most widely-used example being WAMIT (see http:<math>\backslash\backslash</math>www.wamit.com) which determines frequency domain solutions for prescribed structures and structure motions.

==Frequency-domain equation of motion==

The radiation force coefficients also play a significant role in determining the motion of a floating structure possibly subject to mooring and applied forces. To obtain the correct equation of motion in the frequency domain it is important to adopt the approach similar to that taken by [[McIver 2005]], i.e. keep the initial condition terms in the Fourier transform operations. Therefore, if the Fourier transform of the velocity is <math>v_{\mu}(\omega)</math> then the Fourier transform of the acceleration is given by

<center><math>

\int_{0}^{\infty}\ddot{X}_{\mu}(t)e^{i\omega t}dt=-i\omega v_{\mu}(\omega) - V_{\mu}(0)

</math></center>

the time-derivative of the potential obeys a similar relation

<center><math>

\int_{0}^{\infty}\frac{\partial\Phi}{\partial t}e^{i\omega t}dt=-i\omega \phi_{\mu}(\omega) - \Phi(\mathbf{x},0).

</math></center>

Therefore, the ini\partial conditions of the structure <math>(X_{\mu}(0),V_{\mu}(0))</math> and of the potential <math>\Phi(\mathbf{x},0)</math> will be present in the Fourier transform of the equation motion which will be referred to as the frequency-domain equation of motion. If the frequency domain potential is decomposed in the same manner as before then the Fourier transform of the equation of motion for the structure is
<center><math>
M\left[-V_{\mu}(0)-i\omega v_{\mu}(\omega)\right]
=-\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS+F^{S}_{\mu}(\omega)+
i\omega \sum_{\nu} (f_{\mu\nu}(\omega)+i\gamma_{\mu\nu}/\omega)v_{\nu}(\omega)-
\sum_{\nu} c_{\mu\nu}\frac{v(\omega)+X(0)}{-i\omega}+f^{A}_{\mu}(\omega)
</math></center>
for <math>\mu=1,\ldots,6</math>, where <math>f^{A}_{\mu}(\omega)</math> is the Fourier transform of the applied force <math>F_{\mu}(t)</math> in equation~(\ref{linearisedmotion}). Although it is assumed that <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math>, for a non-zero initial velocity <math>\lim_{t^{+}\rightarrow 0}\Phi(\mathbf{x},t)\neq 0</math> because of the impulsive pressure on the free-surface resulting from the initial motion of the pressure. Therefore, the first term on the right-hand side of the above equation is not identically zero, rather is given by
<center><math>
\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS=a(\infty)V(0)
</math></center>
[[Mei 1983]]
where <math>a(\infty)=\lim_{\omega\rightarrow\infty}a(\omega)</math> is the infinite frequency added mass.

The frequency-domain equation is usually re-expressed in the following form
<center><math>
\sum_{\nu} \{c_{\mu\nu}-\omega^{2}\left[M_{\mu\mu}\delta_{\mu\nu}+a_{\mu\nu}(\omega)+i(b_{\mu\nu}+\gamma_{\mu\nu})/\omega\right]\}v_{\nu}(\omega)
= -i\omega\left[ F_{\mu}^{S}(\omega)+f_{\mu}(\omega)+(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)\right]-\sum_{\nu}c_{\mu\nu}X_{\nu}(0)
</math></center>
for <math>\mu=1,\ldots,6</math>. If it is assumed that the structure can only move in one mode of motion and that there is no incident wave or applied force then the complex velocity will be given by
<center><math>
v_{\mu}(\omega)=\frac{-i\omega(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)-c_{\mu\mu}X_{\mu}(0)}{c_{\mu\mu}-\omega^{2}\left[M_{\mu\mu}+a_{\mu\mu}(\omega)+i(b_{\mu\mu}+\gamma_{\mu\mu})/\omega\right]}.
</math></center>

As described by [[McIver 2006]], the zero of the denominator gives the location of the pole referred to as the motion resonance. The complex force coefficient <math>f_{\mu\nu}(\omega)</math> will also possess a pole, however this resonance is annulled by the frequency-dependence of the complex velocity (see [[McIver 2006]]) and so the motion, in the absence of incident waves, is dominated by the motion resonance term. Solving for <math>v(\omega)</math> then the time-domain velocity can be recovered using the inverse transform
<center><math>
V(t)=\frac{1}{\pi} Re \int^{\infty}_{0}v(\omega)e^{-i\omega t} dt.
</math></center>

==Integro-differential equation method==

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by <math>\delta(t-\tau)V_{\alpha}(\tau)</math> where <math>\dot{X_{\alpha}}=V_{\alpha}</math>. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function <math>V_{\alpha}(t)</math> an integral of the impulse response over the range <math>-\infty<\tau<\infty</math> must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

<center><math>(11)

\mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau

</math></center>

where <math>m_{\beta\alpha}(\infty)</math> is an infinite frequency added mass coefficient and <math>L_{\beta\alpha}(t-\tau)</math> is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. <math>V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \}</math>, the impulse response function <math>L_{\beta\alpha}</math> can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

<center><math>

\begin{split}

a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, d\tau \

b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, d\tau

\end{split}

</math></center>

and so if the added mass or damping coefficients are known for all frequencies <math>0\leq\omega\leq\infty</math>, the impulse response or memory function <math>L_{\alpha\beta}(t)</math> can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

<center><math>

\left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t),

</math></center>
for <math>\alpha=1,\ldots,6</math>, where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency <math>\omega</math> on the interval <math>[0,\infty)</math>. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

==Generalised eigenfunction expansion method==

The singularity expansion method (SEM) is based on a general scattering theory for general wave problems and has been adapted by \citeasnoun{hazardSEM} and \citeasnoun{meylaniwwwfb2008}

to problems in hydrodynamics and hydro-elasticity. The central concept in the theory is to express the governing equations for the scattering problem in terms of an evolution equation involving a unitary operator and to use the generalised eigenfunctions and eigenfrequencies of this operator to give an approximation to the time-domain solution of the scattering problem. To give a brief overview of the method the scattering equations must be re-expressed using a harmonic lifting operator <math>\mathbf{B}</math> which maps a function defined on the free-surface onto the total fluid domain. This harmonic lifting operator is denoted by <math>\mathbf{B}</math> so that <math>\mathbf{B}\Psi=\Phi</math> where <math>\Psi=\Phi</math> on the free-surface <math>F</math>. If <math>D</math> denotes the total fluid domain, then potential <math>\Phi</math> satisfies

<center><math>\begin{matrix}

\nabla^{2}\Phi&=0 \quad\textrm{in } D, \

\frac{\partial\Phi}{\partial n} &= 0 \quad\textrm{on } \partial D,\

\Phi&=\Psi \quad\textrm{on } F

\end{matrix}</math></center>

where <math>\partial D</math> includes the bed and the structure surface. The Dirichlet to Neumann map <math>\partial_{\mathbf{n}}\mathbf{B}</math> defined by

<center><math>(12)

\partial_{\mathbf{n}}\mathbf{B}\Psi=\frac{\partial\Phi}{\partial z}, \quad x\in F,

</math></center>

then recovers the corresponding normal derivative to <math>\Psi</math> on the free surface.

The evolution equation

<center><math>

i\frac{\partial \mathbf{U}}{\partial t} = \mathcal{A}\mathbf{U}

</math></center>

accounts for the linear time-dependent free-surface equations using the vector

<center><math>

\mathbf{U}= \begin{pmatrix}

\Psi \

-i\eta

\end{pmatrix},

</math></center>

with <math>\eta</math> denoting the time-dependent free-surface elevation, and the operator <math>\mathcal{A}</math> which is both unitary and self-adjoint given by

<center><math>

\mathcal{A}= \begin{pmatrix}

0 & g \

\partial_{\mathbf{n}}\mathbf{B} & 0 \

\end{pmatrix}

</math></center>

A self-adjoint operator will possess an entirely real spectrum and the generalised eigenfunctions <math>\mathbf{u}=(\psi,-i\zeta)^{T}</math>, which are non-trivial solutions of

<center><math>(13)

\mathcal{A}\mathbf{u}=\omega\mathbf{u},

</math></center>

are just frequency-domain scattering solutions. (On a technical note, the word `generalised' precedes eigenfunction because the energy is unbounded.) This can be shown easily by combining the scalar equations resulting from (13) so as to give the frequency-domain free-surface condition. For a given frequency <math>\omega</math>, the free-surface condition in three-dimensions is satisfied by waves from an infinite number of directions and so for each <math>\omega</math> there is an infinite set of eigenfunctions <math>\mathbf{u}_{n}=(1,\omega/g)^{T}\psi_{n}(\mathbf{x},\omega)</math>, with the <math>n^{th}</math> eigenfunction corresponding to incident waves of the form <math>J_{n}(k r) e^{in\theta}</math>. The general solution of the time-evolution equation is, from spectral theory,

<center><math>(14)

\mathbf{U}(\mathbf{x},t)=\int^{\infty}_{-\infty}\left[ \sum_{n}f_{n}(\omega)\mathbf{u}_{n}(\mathbf{x},\omega) \right]e^{-i\omega t}\, d\omega

</math></center>

where <math>f_{n}(\omega)</math> is determined by the initial conditions now expressed as <math>\mathbf{U}(\mathbf{x},0)</math>. To obtain this expression for <math>f_{n}(\omega)</math>, apply the energy inner product to (14) evaluated at <math>t=0</math>. The eigenfunctions satisfy the orthogonality condition

<center><math>(15)

<\mathbf{u}_{m}(\mathbf{x},\omega),\mathbf{u}_{n}(\mathbf{x},\omega')>_{E}=\Lambda_{m}(\omega)\delta_{mn}\delta(\omega-\omega')

</math></center>

where <math><>_{E}</math> denotes a special energy inner product (defined with a <math>\mathcal{H}</math> subscript by \citeasnoun{meylaniwwwfb2008}) and it can be shown that

<center><math>

\Lambda_{m}(\omega)=\frac{4\pi\omega^{2}}{g k}\frac{d\omega}{dk}.

</math></center>

Therefore, it is straightforward to show that

<center><math>

f_{n}(\omega)=\frac{1}{\Lambda_{n}(\omega)}<\mathbf{U}(\mathbf{x},0),\mathbf{u}_{n}(\mathbf{x},\omega)>_{E}.

</math></center>

by evaluating the energy inner product <math><\mathbf{U}(\mathbf{x},0),\mathbf{u}_{n}(\mathbf{x},\omega)>_{E}</math> using the orthogonality relation~(15) and the definition~(14). This expression can be further simplified using the definition of the energy product but details will not be provided here. Instead, it should be noted that an analytic expression for the general solution has been obtained. Thus, for a given scattering problem it is in theory possible to obtain the motion of the free-surface and the potential on the free-surface.

The singularity expansion method involves moving the path of integration in~(14) in the <math>\omega</math>-plane across the singularities in the lower half plane Im<math>\,\omega\leq0</math> using the method of contour integration. Thus, the general solution will consist of contributions from the poles, any branch cuts, the path at infinity and the remainder of the path located below the poles in the <math>\omega</math>-plane. As explained by \citeasnoun{hazardSEM}, the contributions from infinity are assumed to be identically zero and the contribution from the remainder of the path decays faster than the pole contributions. Given that the contribution from the branch cuts are thought to be significant only for very large times an approximation to the general solution can be obtained for medium and large <math>t</math> because the contributions from the poles dominate for this range of times. Therefore, the SEM requires a knowledge of the pole structure of the integrand to approximate the complete integral. Furthermore, the behaviour of the eigenfunctions in the vicinity of the poles must be known in order to compute the residue of each pole. The scattering potential will satisfy

<center><math>

(I+T(\omega))\phi^{S}=g_{I}

</math></center>

where the form of the operator <math>T(\omega)</math> depends on the solution method chosen and <math>g_{I}</math> is determined by the incident wave. Given <math>g_{I}</math>, the scattered field is

<center><math>

\phi^{S}=(I+T(\omega))^{-1}g_{I}

</math></center>

and the pole structure of <math>\phi_{S}</math> will be inherited from the operator <math>(I+T(\omega))^{-1}</math>, referred to as the resolvent.

In the case where the resolvent operator is approximated by a matrix, it can be shown that the potential has the form

<center><math>(16)

\phi_{n}(\mathbf{x},\omega)\sim\frac{\alpha_{nj}v_{j}(\mathbf{x})}{\omega-\omega_{j}} \textrm{ as } \omega\rightarrow\omega_{j}

</math></center>

in the vicinity of the pole. The poles themselves are determined by locating values of <math>\omega</math> in the lower complex plane where the resolvent is not invertible. Thus, <math>v_{j}</math> is a generalised eigenfunction of <math>A_{j0}=I+T(\omega_{j})</math>. The integral in equation~(14) can be approximated by closing the integration path in the lower half plane and then moving the integration path across a finite number of <math>P</math> poles and summing over the contributions of these poles. As described by~\citeasnoun{hazardSEM}, any branch cuts in the complex <math>\omega</math>-plane will only be significant at very large times and the integrals at infinity are expected to be zero. Therefore, the contribution of the <math>P</math> poles closest to the real-<math>\omega</math> axis (and hence with the smallest decay rates <math>e^{-Im(\omega_{j})}</math>) will dominate in the medium term. So, after neglecting other contributions to the integral and using~(16) to determine the residues of the poles, the potential on the free-surface will be given by

<center><math>(17)

\Psi(\mathbf{x},t)\approx -2\pi\sum_{j}\left[ \sum_{n}f_{n}(\omega_{j})\alpha_{nj}\right]v_{j}(\mathbf{x},0)e^{-i\omega_{j}t}.

</math></center>

Although full details of how this expression can be evaluated are not given here, examples of applications of this method are given by~\citeasnoun{meylaniwwwfb2007}, \citeasnoun{meylaniwwwfb2008} and \citeasnoun{meylan2002}. Each of these papers also contain more detailed instructions regarding the computational aspects of the method, such as determining the locations of the poles.

\citeasnoun{hazardSEM} notes that the SEM is a non-rigorous method and requires some heuristic arguments to justify its use. However, in the specific cases considered the results are generally accurate for a large range of times and much frequency-domain information regarding resonances is inherent in the time-domain solution. At present, the method only applies to scattering problems; however, it is hoped that it can be extended to radiation and coupled motion problems. It is not clear how the forces on the structure will be determined.

\f1

}

{\rtf1\ansi\ansicpg1252\deff0\deflang2057{\fonttbl{\f0\fnil\fcharset0 Courier New;}{\f1\fswiss\fcharset0 Arial;}}

{\*\generator Msftedit 5.41.21.2508;}\viewkind4\uc1d\f0\fs20=Review of time-domain models=

==Frequency domain definitions==(1)

Linearised water-wave problems were first investigated in the frequency domain wherein all motions are assumed to be time-harmonic and to have begun from <math>t= -\infty</math> . The time-dependence can be removed in a straightforward manner by substituting for the potential and any structural motions using

<center><math>(2)

\begin{split}

\Phi(\mbf{x},t)=Re \{\phi(\mbf{x},\omega) e^{-i\omega t}\} \

V(\mbf{x},t)=Re \{ v(\mbf{x},\omega) e^{-i\omega t}\}

\end{split}

</math></center>

and this results in a considerable simplification of the problem. The complex amplitudes of the dynamic quantities vary with frequency and by removing the time-dependence the governing equations and boundary conditions will feature these terms only.
However, \citeasnoun{mciver2003} have shown that the frequency-domain potential can also be defined using the following operations on the time-domain solution:

*take a Laplace transform of the time-domain potential
<center><math>
\hat{\phi}(\mbf{x},s)=\int^{\infty}_{0}\Phi(\mbf{x},t)e^{-s t}\, dt ,\quad \textrm{Re } s >0;
</math></center>

*apply the change of variables <math>s=-i\omega</math> so that
<center><math>
\phi(\mbf{x},\omega)=\hat{\phi}(\mbf{x},-i\omega).
</math></center>

*noting that <math>\phi(\mbf{x},-\omega)=\bar{\phi}(\mbf{x},\omega)</math>, the inverse Fourier transform is given by
<center><math>(3)
\Phi(\mbf{x},t)=\frac{1}{2\pi}\int^{\infty}_{-\infty}\phi(\mbf{x},\omega)e^{-i\omega t} d\omega =\frac{1}{\pi}Re\int^{\infty}_{0}\phi(\mbf{x},\omega)e^{-i\omega t}\,d\omega
</math></center>
where the path of integration must pass over any singularities of <math>\phi</math> that lie on the real axis and it has been assumed that there is no motion prior to <math>t=0</math>, i.e. <math>\Phi(\mbf{x},t)=0</math> for <math>t<0</math>.

If <math>\omega</math> is real then <math>\phi(\mbf{x},\omega)</math> is just the frequency domain potential. By the principle of causality, which precludes the existence of the response before the cause so that <math>\Phi(\mbf{x},t)=0</math> for <math>t<0</math> , no singularities can exist in <math>\textrm{Im } \omega>0</math>. Therefore, if it is assumed that no singularities exist on the real axis the inverse transform will integrate the frequency-domain potential over the full range of <math>\omega</math>. With this approach, the frequency-domain equations are obtained by applying the Fourier transform to the time-dependent equations. However, unlike the equations resulting from the assumptions~(2) of time-harmonic motion from <math>t= -\infty</math>, the initial conditions will be present in the frequency-domain equations arising from the Fourier transformation. Although, the latter equations are more mathematically consistent it is usual to set all initial conditions to zero before attempting to solve the equations even if this is not the case. Nevertheless, in the frequency-domain analysis of resonances in coupled motion problems \citeasnoun{pmciver2005} retains the initial condition terms in the equation of motion for the structure and obtains some important results regarding the resonant behaviour of the motion. Therefore, the retention of the initial condition terms depends on the application of the subsequent equations.

Many analytical and approximate solutions exist for frequency-domain problems involving structures with simple geometries. Thus, in the simplest frequency-domain radiation and diffraction problems it is possible to completely determine the velocity potential analytically. The hydrodynamic forces, which are crucial in the context of marine engineering, can then be calculated directly from the potential. For more general geometries this is not possible and the hydrodynamic forces must be computed using numerical methods. Nevertheless, it is useful to consider the analytical expressions for the various potentials and forces in the frequency-domain problem.

In a linearised coupled motion problem, it is customary to decompose the total velocity potential into a scattering potential <math>\phi^{S}</math> and a radiation potential <math>\phi^{R}</math>. Thus, the coupled motion problem will require the solution of the radiation problem and the scattering problem. The radiation potential can be further decomposed into a sum over the six modes of motion so that the total frequency domain potential is

<center><math>(4)

\phi=\phi^{S}+\sum_{\mu}v_{\mu}\phi_{\mu}

</math></center>

where <math>u_{\mu}</math> is the complex amplitude of the generalised velocity in the <math>\mu</math> direction and <math>\phi_{\mu}</math> describes the fluid response due to the forced oscillations in mode <math>\mu</math> with unit velocity amplitude. The velocities in each mode will be determined from the frequency domain equation of motion for the body. However, this requires the computation of the radiation and exciting forces on the structure. Therefore, it is necessary to consider the radiation problem prior to the coupled motion equation. It is first important to note that, for a given mode <math>\mu</math> the boundary condition on the structure will be

<center><math>

\frac{tial\phi_{\mu}}{tial n}=n_{\mu},

</math></center>

because the total velocity is

<center><math>

v(\omega)=\sum_{\mu}v_{\mu}n_{\mu}

</math></center>

where <math>n_{\mu}</math> is the <math>\mu</math> component of the generalised normal and <math>v_{\mu}</math> is the component of the generalised complex velocity amplitude in this direction. The hydrodynamic force on the structure in the <math>\mu</math> direction due to the fluid response to the forced oscillations is

<center><math>(5)

F_{\mu}^{R}=i\omega\rho\iint_{S_{B}}\phi^{R}n_{\mu} \,dS

</math></center>

and the force due to the diffraction of an incident wave by the fixed structure is

<center><math>(6)

F_{\mu}^{S}=i\omega\rho\iint_{S_{B}}\phi^{S}n_{\mu} \,dS

</math></center>

where the time-dependence has been removed in both cases. By expressing the radiation potential as a sum over the individual modes, the radiation force can written as <math>\sum_{\nu} v_{\nu}f_{\nu\mu}</math> where

<center><math>

f_{\nu\mu}=i\omega\rho\iint_{\Gamma} \phi_{\nu}n_{\mu} \,dS.

</math></center>

It is conventional to decompose the radiation force into a term featuring the added mass matrix <math>a_{\alpha\beta}</math> and a term featuring the damping matrix <math>b_{\alpha\beta}</math> as follows

<center><math>(7)

f_{\beta\alpha}=i\omega(a_{\beta\alpha}+\frac{ib_{\beta\alpha}}{\omega})

</math></center>

where the added mass and damping coefficients are in phase with the acceleration and velocity respectively. These real quantities depend only on the geometry of the structure and the oscillation frequency <math>\omega</math> and describe many of the properties of the structure. In particular, (as will be shown see later) if the variation of these coefficients with frequency is known then we can predict the resonant behaviour of the structure in the time-domain as well as the frequency domain. The computation of these quantities was considered to be very important in marine engineering and naval architecture and so a variety of different computer codes were developed to compute the coefficients efficiently. The boundary element method forms the basis of most of the computer code algorithms, the most widely-used example being WAMIT (see http:<math>\backslash\backslash</math>www.wamit.com) which determines frequency domain solutions for prescribed structures and structure motions.

==Frequency-domain equation of motion==(8)

The radiation force coefficients also play a significant role in determining the motion of a floating structure possibly subject to mooring and applied forces. To obtain the correct equation of motion in the frequency domain it is important to adopt the approach similar to that taken by~\citeasnoun{pmciver2005}, i.e. keep the initial condition terms in the Fourier transform operations. Therefore, if the Fourier transform of the velocity is <math>v_{\mu}(\omega)</math> then the Fourier transform of the acceleration is given by

<center><math>

\int_{0}^{\infty}\ddot{X}_{\mu}(t)e^{i\omega t}dt=-i\omega v_{\mu}(\omega) - V_{\mu}(0)

</math></center>

the time-derivative of the potential obeys a similar relation

<center><math>

\int_{0}^{\infty}\frac{tial\Phi}{tial t}e^{i\omega t}dt=-i\omega \phi_{\mu}(\omega) - \Phi(\mbf{x},0).

</math></center>

Therefore, the initial conditions of the structure <math>(X_{\mu}(0),V_{\mu}(0))</math> and of the potential <math>\Phi(\mbf{x},0)</math> will be present in the Fourier transform of the equation motion which will be referred to as the frequency-domain equation of motion. If the frequency domain potential is decomposed in the same manner as before~(4) then the Fourier transform of the equation of motion~(\ref{linearisedmotion}) for the structure is

<center><math>(9)

\begin{split}

M\left[-V_{\mu}(0)-i\omega v_{\mu}(\omega)\right]&

=-\rho\iint_{\Gamma}(-\Phi(\mbf{x},0))n_{\mu}dS+F^{S}_{\mu}(\omega)+\

i\omega \sum_{\nu}&(f_{\mu\nu}(\omega)+i\gamma_{\mu\nu}/\omega)v_{\nu}(\omega)-

\sum_{\nu} c_{\mu\nu}\frac{v(\omega)+X(0)}{-i\omega}+f^{A}_{\mu}(\omega)

\end{split}

</math></center>

for <math>\mu=1,\ldots,6</math>, where <math>f^{A}_{\mu}(\omega)</math> is the Fourier transform of the applied force <math>F_{\mu}(t)</math> in equation~(\ref{linearisedmotion}). Although it is assumed that <math>\Phi(\mbf{x},t)=0</math> for <math>t<0</math>, for a non-zero initial velocity <math>\lim_{t^{+}\rightarrow 0}\Phi(\mbf{x},t)\neq 0</math> because of the impulsive pressure on the free-surface resulting from the initial motion of the pressure. Therefore, the first term on the right-hand side of the above equation is not identically zero, rather is given by

<center><math>

\rho\iint_{\Gamma}(-\Phi(\mbf{x},0))n_{\mu}dS=a(\infty)V(0)

</math></center>

(see \citeasnoun{ccmei2}, \S~8.12.1)

where <math>a(\infty)=\lim_{\omega\rightarrow\infty}a(\omega)</math> is the infinite frequency added mass.

The frequency-domain equation is usually re-expressed in the following form

<center><math>

\begin{split}

\sum_{\nu}&\{c_{\mu\nu}-\omega^{2}\left[M_{\mu\mu}\delta_{\mu\nu}+a_{\mu\nu}(\omega)+i(b_{\mu\nu}+\gamma_{\mu\nu})/\omega\right]\}v_{\nu}(\omega) \

=& -i\omega\left[ F_{\mu}^{S}(\omega)+f_{\mu}(\omega)+(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)\right]-\sum_{\nu}c_{\mu\nu}X_{\nu}(0)\end{split}

</math></center>

for <math>\mu=1,\ldots,6</math>. If it is assumed that the structure can only move in one mode of motion and that there is no incident wave or applied force then the complex velocity will be given by

<center><math>(10)

v_{\mu}(\omega)=\frac{-i\omega(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)-c_{\mu\mu}X_{\mu}(0)}{c_{\mu\mu}-\omega^{2}\left[M_{\mu\mu}+a_{\mu\mu}(\omega)+i(b_{\mu\mu}+\gamma_{\mu\mu})/\omega\right]}.

</math></center>

As described by \citeasnoun{mciver2006}, the zero of the denominator gives the location of the pole referred to as the motion resonance. The complex force coefficient <math>f_{\mu\nu}(\omega)</math> will also possess a pole, however this resonance is annulled by the frequency-dependence of the complex velocity (see \S~6 of \citeasnoun{mciver2006}) and so the motion, in the absence of incident waves, is dominated by the motion resonance term. Solving equation~(10) for <math>v(\omega)</math> then the time-domain velocity can be recovered using the inverse transform

<center><math>

V(t)=\frac{1}{\pi} Re \int^{\infty}_{0}v(\omega)e^{-i\omega t} dt.

</math></center>

==Integro-differential equation method==

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by <math>\delta(t-\tau)V_{\alpha}(\tau)</math> where <math>\dot{X_{\alpha}}=V_{\alpha}</math>. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function <math>V_{\alpha}(t)</math> an integral of the impulse response over the range <math>-\infty<\tau<\infty</math> must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

<center><math>(11)

\mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau

</math></center>

where <math>m_{\beta\alpha}(\infty)</math> is an infinite frequency added mass coefficient and <math>L_{\beta\alpha}(t-\tau)</math> is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. <math>V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \}</math>, the impulse response function <math>L_{\beta\alpha}</math> can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

<center><math>

\begin{split}

a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, d\tau \

b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, d\tau

\end{split}

</math></center>

and so if the added mass or damping coefficients are known for all frequencies <math>0\leq\omega\leq\infty</math>, the impulse response or memory function <math>L_{\alpha\beta}(t)</math> can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

<center><math>

\left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t),

</math></center>
for <math>\alpha=1,\ldots,6</math>, where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency <math>\omega</math> on the interval <math>[0,\infty)</math>. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

==Generalised eigenfunction expansion method==

The singularity expansion method (SEM) is based on a general scattering theory for general wave problems and has been adapted by \citeasnoun{hazardSEM} and \citeasnoun{meylaniwwwfb2008}

to problems in hydrodynamics and hydro-elasticity. The central concept in the theory is to express the governing equations for the scattering problem in terms of an evolution equation involving a unitary operator and to use the generalised eigenfunctions and eigenfrequencies of this operator to give an approximation to the time-domain solution of the scattering problem. To give a brief overview of the method the scattering equations must be re-expressed using a harmonic lifting operator <math>\mbf{B}</math> which maps a function defined on the free-surface onto the total fluid domain. This harmonic lifting operator is denoted by <math>\mbf{B}</math> so that <math>\mbf{B}\Psi=\Phi</math> where <math>\Psi=\Phi</math> on the free-surface <math>F</math>. If <math>D</math> denotes the total fluid domain, then potential <math>\Phi</math> satisfies

<center><math>\begin{matrix}

\nabla^{2}\Phi&=0 \quad\textrm{in } D, \

\frac{tial\Phi}{tial n} &= 0 \quad\textrm{on } tial D,\

\Phi&=\Psi \quad\textrm{on } F

\end{matrix}</math></center>

where <math>tial D</math> includes the bed and the structure surface. The Dirichlet to Neumann map <math>tial_{\mbf{n}}\mbf{B}</math> defined by

<center><math>(12)

tial_{\mbf{n}}\mbf{B}\Psi=\frac{tial\Phi}{tial z}, \quad x\in F,

</math></center>

then recovers the corresponding normal derivative to <math>\Psi</math> on the free surface.

The evolution equation

<center><math>

i\frac{tial \mbf{U}}{tial t} = \mathcal{A}\mbf{U}

</math></center>

accounts for the linear time-dependent free-surface equations using the vector

<center><math>

\mbf{U}= \begin{pmatrix}

\tab \Psi \

\tab -i\eta

\end{pmatrix},

</math></center>

with <math>\eta</math> denoting the time-dependent free-surface elevation, and the operator <math>\mathcal{A}</math> which is both unitary and self-adjoint given by

<center><math>

[[Category:Linear Water-Wave Theory]]

{{complete page}}

Category:Time-Dependent Linear Water Waves

2009-08-20T23:06:46Z

Cfitzgerald: /* Time domain problem for a floating structure */

Generally the focus of research is on the [[Frequency Domain Problem]]. The time-domain
problem can be solved by [[Generalised Eigenfunction Expansion]] or by
an [[Expansion over the Resonances]] or using [[Memory Effect Function]].

== Introduction ==

Analytical solutions of the linearised time-domain equations for a coupled-motion or forced-motion problems are very rare. In fact, only a small number of analytic time-domain solutions have been obtained (see the papers by [[Kennard 1949]] and [[McIver 1994]]) for structures with simple geometries. Therefore, a variety of numerical methods have been developed to solve initial-value time-domain wave-structure interaction problems. The Fourier transform enables the time and frequency domain quantities to be related and the frequency domain forces and potentials have important roles in the development of the numerical time-domain solution methods. This can be partly attributed to the fact that, in the past, analytical or semi-analytical solutions to linearised water-wave problems were more easily obtained in the frequency domain given that the assumption of time-harmonic motion results in a significant simplification of the interaction equations. Therefore, it was natural to develop time-domain solutions based on frequency domain results and the Fourier transform relation between the domains.

== Time domain problem for a floating structure ==
To begin with, it is useful to introduce the general time-dependent water wave problem. Therefore, the the time-domain equations for the motion of a floating structure must be quoted. From these equations, the forced motion and fixed body interaction problems (i.e. the radiation and scattering problems) can be derived. Furthermore, the corresponding frequency-domain equations can be derived from the time-domain in order to illustrate the important relation between the two. For a semi-submerged, vertically axisymmetric structure the time-domain linearised equations are
<center><math>
\Delta\Phi\left( \mathbf{x,}t\right) =0,\ \ \mathbf{x}\in\Omega,
</math></center>
<center><math>
\partial_{n}\Phi=0,\ \ z=-h,
</math></center>
<center><math>
\partial_{n}\Phi=\sum\dot{X}_{\nu}n_{\nu},\ \ mathbf{x}\in\Gamma,
</math></center>
where <math>\Phi</math> is the velocity potential for the fluid, <math>\Gamma</math> is the still water position of the wetted surface of the structure. At the free surface <math>F</math>, the potential must obey the kinematic condition
<center><math>
\partial_{t}\zeta=\partial_{n}\Phi,\ \ z=0,\ x\in F,
</math></center>
and the dynamic condition (the linearized Bernoulli equation)
<center><math>
\zeta = -(1/g)\partial_{t}\Phi,\ \ z=0,\ x\in F,
</math></center>
where <math>\zeta</math> is the free-surface elevation. Initial conditions for <math>\Phi</math> and <math>\partial_{t}\Phi</math> must also be specified. These fluid motion equations must be combined with the equation of motion of the body to give the coupled motion problem involving the structure motions, described by the displacement <math>X_{\nu}</math> in mode <math> \nu</math>, and the velocity potential <math>\Phi</math>. The equation of motion for a structure moored by an arrangement of linear springs and dampers is
<center><math>
M_{\mu\mu}\ddot{X}_{\mu}=-\rho\iint_{\Gamma} \frac{\partial\Phi}{\partial t} n_{\mu}\, dS-\sum^{6}_{\nu=1}\left[ c_{\mu\nu}X_{\nu}(t)+\gamma_{\mu\nu}\dot{X}_{\nu}\right]+F_{\mu}(t), \qquad \mu=1,\ldots,6
</math></center>
where <math>M_{\mu\mu}</math> are the diagonal elements of mass matrix for the structure, i.e. <math>M_{\mu\mu}=M</math> for <math>\mu=1,2,3</math> and <math>M_{\mu\mu}=I_{\mu-3,\mu-3}</math> for <math>\mu=4,5,6</math>.The characteristics of the mooring springs and dampers are described the matrices <math>\kappa_{\mu\nu}</math> and <math>\gamma_{\mu\nu}</math> , with the spring term included in the term <math>c_{\mu\nu}=\rho g b_{\mu}\delta_{\mu\nu}+k_{\mu\nu}</math> which also describes the effect of buoyancy. The initial generalised displacements <math>X_{\mu}</math> and velocities <math>\dot{X}_{\mu}</math> of the body must be specified for all modes <math>\mu=1,\ldots,6</math> in order to solve the equation.
It is non-zero for the heave, roll and pitch modes only and these terms are <math>b_{3}=W</math>, <math>b_{4}=I^{A}_{22}+I^{V}_{3}</math> and <math>b_{5}=I^{A}_{11}+I^{V}_{3}</math> respectively, where <math>W</math> is the waterplane area and <math>I^{A}</math> is the moment of the waterplane are (see Chapter 7, Mei (1983)) and <math>I^{V}_{3}</math> is <math>z</math>-component centre of buoyancy of the structure. Thus, the coupled equations of motion for a floating structure subjected to some constraint forces have been derived. (N.B. it is assumed that the centre of rotation and the centre of mass of the structure coincide for this equation, i.e. it is assumed that the body is semi-submerged. Furthermore, any wave incidence is assumed to be along the <math>x</math>-axis.) The scattering and radiation problems are simpler than the coupled problem because the motion of the the structure is then prescribed.

{{Equations for fixed bodies in the time domain}}

==Frequency domain definitions==

Linearised water-wave problems were first investigated in the frequency domain wherein all motions are assumed to be time-harmonic and to have begun from <math>t= -\infty</math> . The time-dependence can be removed in a straightforward manner by substituting for the potential and any structural motions using
<center><math>
\Phi(\mathbf{x},t)=Re \{\phi(\mathbf{x},\omega) e^{-i\omega t}\}
</math></center>
<center><math>
V(\mathbf{x},t)=Re \{ v(\mathbf{x},\omega) e^{-i\omega t}\}
</math></center>
and this results in a considerable simplification of the problem. The complex amplitudes of the dynamic quantities vary with frequency and by removing the time-dependence the governing equations and boundary conditions will feature these terms only.
However, \citeasnoun{mciver2003} have shown that the frequency-domain potential can also be defined using the following operations on the time-domain solution:

*take a Laplace transform of the time-domain potential
<center><math>
\hat{\phi}(\mathbf{x},s)=\int^{\infty}_{0}\Phi(\mathbf{x},t)e^{-s t}\, dt ,\quad \textrm{Re } s >0;
</math></center>

*apply the change of variables <math>s=-i\omega</math> so that
<center><math>
\phi(\mathbf{x},\omega)=\hat{\phi}(\mathbf{x},-i\omega).
</math></center>

*noting that <math>\phi(\mathbf{x},-\omega)=\bar{\phi}(\mathbf{x},\omega)</math>, the inverse Fourier transform is given by
<center><math>(3)
\Phi(\mathbf{x},t)=\frac{1}{2\pi}\int^{\infty}_{-\infty}\phi(\mathbf{x},\omega)e^{-i\omega t} d\omega =\frac{1}{\pi}Re\int^{\infty}_{0}\phi(\mathbf{x},\omega)e^{-i\omega t}\,d\omega
</math></center>
where the path of integration must pass over any singularities of <math>\phi</math> that lie on the real axis and it has been assumed that there is no motion prior to <math>t=0</math>, i.e. <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math>.

If <math>\omega</math> is real then <math>\phi(\mathbf{x},\omega)</math> is just the frequency domain potential. By the principle of causality, which precludes the existence of the response before the cause so that <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math> , no singularities can exist in <math>\textrm{Im } \omega>0</math>. Therefore, if it is assumed that no singularities exist on the real axis the inverse transform will integrate the frequency-domain potential over the full range of <math>\omega</math>. With this approach, the frequency-domain equations are obtained by applying the Fourier transform to the time-dependent equations. However, unlike the equations resulting from the assumptions~(2) of time-harmonic motion from <math>t= -\infty</math>, the initial conditions will be present in the frequency-domain equations arising from the Fourier transformation. Although, the latter equations are more mathematically consistent it is usual to set all initial conditions to zero before attempting to solve the equations even if this is not the case. Nevertheless, in the frequency-domain analysis of resonances in coupled motion problems \citeasnoun{pmciver2005} retains the initial condition terms in the equation of motion for the structure and obtains some important results regarding the resonant behaviour of the motion. Therefore, the retention of the initial condition terms depends on the application of the subsequent equations.

Many analytical and approximate solutions exist for frequency-domain problems involving structures with simple geometries. Thus, in the simplest frequency-domain radiation and diffraction problems it is possible to completely determine the velocity potential analytically. The hydrodynamic forces, which are crucial in the context of marine engineering, can then be calculated directly from the potential. For more general geometries this is not possible and the hydrodynamic forces must be computed using numerical methods. Nevertheless, it is useful to consider the analytical expressions for the various potentials and forces in the frequency-domain problem.

In a linearised coupled motion problem, it is customary to decompose the total velocity potential into a scattering potential <math>\phi^{S}</math> and a radiation potential <math>\phi^{R}</math>. Thus, the coupled motion problem will require the solution of the radiation problem and the scattering problem. The radiation potential can be further decomposed into a sum over the six modes of motion so that the total frequency domain potential is
<center><math>
\phi=\phi^{S}+\sum_{\mu}v_{\mu}\phi_{\mu}
</math></center>

where <math>u_{\mu}</math> is the complex amplitude of the generalised velocity in the <math>\mu</math> direction and <math>\phi_{\mu}</math> describes the fluid response due to the forced oscillations in mode <math>\mu</math> with unit velocity amplitude. The velocities in each mode will be determined from the frequency domain equation of motion for the body. However, this requires the computation of the radiation and exciting forces on the structure. Therefore, it is necessary to consider the radiation problem prior to the coupled motion equation. It is first important to note that, for a given mode <math>\mu</math> the boundary condition on the structure will be

<center><math>

\frac{\partial\phi_{\mu}}{\partial n}=n_{\mu},

</math></center>

because the total velocity is

<center><math>

v(\omega)=\sum_{\mu}v_{\mu}n_{\mu}

</math></center>

where <math>n_{\mu}</math> is the <math>\mu</math> component of the generalised normal and <math>v_{\mu}</math> is the component of the generalised complex velocity amplitude in this direction. The hydrodynamic force on the structure in the <math>\mu</math> direction due to the fluid response to the forced oscillations is

<center><math>(5)

F_{\mu}^{R}=i\omega\rho\iint_{S_{B}}\phi^{R}n_{\mu} \,dS

</math></center>

and the force due to the diffraction of an incident wave by the fixed structure is

<center><math>(6)

F_{\mu}^{S}=i\omega\rho\iint_{S_{B}}\phi^{S}n_{\mu} \,dS

</math></center>

where the time-dependence has been removed in both cases. By expressing the radiation potential as a sum over the individual modes, the radiation force can written as <math>\sum_{\nu} v_{\nu}f_{\nu\mu}</math> where

<center><math>

f_{\nu\mu}=i\omega\rho\iint_{\Gamma} \phi_{\nu}n_{\mu} \,dS.

</math></center>

It is conventional to decompose the radiation force into a term featuring the added mass matrix <math>a_{\alpha\beta}</math> and a term featuring the damping matrix <math>b_{\alpha\beta}</math> as follows

<center><math>(7)

f_{\beta\alpha}=i\omega(a_{\beta\alpha}+\frac{ib_{\beta\alpha}}{\omega})

</math></center>

where the added mass and damping coefficients are in phase with the acceleration and velocity respectively. These real quantities depend only on the geometry of the structure and the oscillation frequency <math>\omega</math> and describe many of the properties of the structure. In particular, (as will be shown see later) if the variation of these coefficients with frequency is known then we can predict the resonant behaviour of the structure in the time-domain as well as the frequency domain. The computation of these quantities was considered to be very important in marine engineering and naval architecture and so a variety of different computer codes were developed to compute the coefficients efficiently. The boundary element method forms the basis of most of the computer code algorithms, the most widely-used example being WAMIT (see http:<math>\backslash\backslash</math>www.wamit.com) which determines frequency domain solutions for prescribed structures and structure motions.

==Frequency-domain equation of motion==

The radiation force coefficients also play a significant role in determining the motion of a floating structure possibly subject to mooring and applied forces. To obtain the correct equation of motion in the frequency domain it is important to adopt the approach similar to that taken by [[McIver 2005]], i.e. keep the initial condition terms in the Fourier transform operations. Therefore, if the Fourier transform of the velocity is <math>v_{\mu}(\omega)</math> then the Fourier transform of the acceleration is given by

<center><math>

\int_{0}^{\infty}\ddot{X}_{\mu}(t)e^{i\omega t}dt=-i\omega v_{\mu}(\omega) - V_{\mu}(0)

</math></center>

the time-derivative of the potential obeys a similar relation

<center><math>

\int_{0}^{\infty}\frac{\partial\Phi}{\partial t}e^{i\omega t}dt=-i\omega \phi_{\mu}(\omega) - \Phi(\mathbf{x},0).

</math></center>

Therefore, the ini\partial conditions of the structure <math>(X_{\mu}(0),V_{\mu}(0))</math> and of the potential <math>\Phi(\mathbf{x},0)</math> will be present in the Fourier transform of the equation motion which will be referred to as the frequency-domain equation of motion. If the frequency domain potential is decomposed in the same manner as before then the Fourier transform of the equation of motion for the structure is
<center><math>
M\left[-V_{\mu}(0)-i\omega v_{\mu}(\omega)\right]
=-\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS+F^{S}_{\mu}(\omega)+
i\omega \sum_{\nu} (f_{\mu\nu}(\omega)+i\gamma_{\mu\nu}/\omega)v_{\nu}(\omega)-
\sum_{\nu} c_{\mu\nu}\frac{v(\omega)+X(0)}{-i\omega}+f^{A}_{\mu}(\omega)
</math></center>
for <math>\mu=1,\ldots,6</math>, where <math>f^{A}_{\mu}(\omega)</math> is the Fourier transform of the applied force <math>F_{\mu}(t)</math> in equation~(\ref{linearisedmotion}). Although it is assumed that <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math>, for a non-zero initial velocity <math>\lim_{t^{+}\rightarrow 0}\Phi(\mathbf{x},t)\neq 0</math> because of the impulsive pressure on the free-surface resulting from the initial motion of the pressure. Therefore, the first term on the right-hand side of the above equation is not identically zero, rather is given by
<center><math>
\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS=a(\infty)V(0)
</math></center>
[[Mei 1983]]
where <math>a(\infty)=\lim_{\omega\rightarrow\infty}a(\omega)</math> is the infinite frequency added mass.

The frequency-domain equation is usually re-expressed in the following form
<center><math>
\sum_{\nu} \{c_{\mu\nu}-\omega^{2}\left[M_{\mu\mu}\delta_{\mu\nu}+a_{\mu\nu}(\omega)+i(b_{\mu\nu}+\gamma_{\mu\nu})/\omega\right]\}v_{\nu}(\omega)
= -i\omega\left[ F_{\mu}^{S}(\omega)+f_{\mu}(\omega)+(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)\right]-\sum_{\nu}c_{\mu\nu}X_{\nu}(0)
</math></center>
for <math>\mu=1,\ldots,6</math>. If it is assumed that the structure can only move in one mode of motion and that there is no incident wave or applied force then the complex velocity will be given by
<center><math>
v_{\mu}(\omega)=\frac{-i\omega(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)-c_{\mu\mu}X_{\mu}(0)}{c_{\mu\mu}-\omega^{2}\left[M_{\mu\mu}+a_{\mu\mu}(\omega)+i(b_{\mu\mu}+\gamma_{\mu\mu})/\omega\right]}.
</math></center>

As described by [[McIver 2006]], the zero of the denominator gives the location of the pole referred to as the motion resonance. The complex force coefficient <math>f_{\mu\nu}(\omega)</math> will also possess a pole, however this resonance is annulled by the frequency-dependence of the complex velocity (see [[McIver 2006]]) and so the motion, in the absence of incident waves, is dominated by the motion resonance term. Solving for <math>v(\omega)</math> then the time-domain velocity can be recovered using the inverse transform
<center><math>
V(t)=\frac{1}{\pi} Re \int^{\infty}_{0}v(\omega)e^{-i\omega t} dt.
</math></center>

==Integro-differential equation method==

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by <math>\delta(t-\tau)V_{\alpha}(\tau)</math> where <math>\dot{X_{\alpha}}=V_{\alpha}</math>. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function <math>V_{\alpha}(t)</math> an integral of the impulse response over the range <math>-\infty<\tau<\infty</math> must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

<center><math>(11)

\mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau

</math></center>

where <math>m_{\beta\alpha}(\infty)</math> is an infinite frequency added mass coefficient and <math>L_{\beta\alpha}(t-\tau)</math> is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. <math>V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \}</math>, the impulse response function <math>L_{\beta\alpha}</math> can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

<center><math>

\begin{split}

a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, d\tau \

b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, d\tau

\end{split}

</math></center>

and so if the added mass or damping coefficients are known for all frequencies <math>0\leq\omega\leq\infty</math>, the impulse response or memory function <math>L_{\alpha\beta}(t)</math> can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

<center><math>

\left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t),

</math></center>
for <math>\alpha=1,\ldots,6</math>, where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency <math>\omega</math> on the interval <math>[0,\infty)</math>. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

==Generalised eigenfunction expansion method==

The singularity expansion method (SEM) is based on a general scattering theory for general wave problems and has been adapted by \citeasnoun{hazardSEM} and \citeasnoun{meylaniwwwfb2008}

to problems in hydrodynamics and hydro-elasticity. The central concept in the theory is to express the governing equations for the scattering problem in terms of an evolution equation involving a unitary operator and to use the generalised eigenfunctions and eigenfrequencies of this operator to give an approximation to the time-domain solution of the scattering problem. To give a brief overview of the method the scattering equations must be re-expressed using a harmonic lifting operator <math>\mathbf{B}</math> which maps a function defined on the free-surface onto the total fluid domain. This harmonic lifting operator is denoted by <math>\mathbf{B}</math> so that <math>\mathbf{B}\Psi=\Phi</math> where <math>\Psi=\Phi</math> on the free-surface <math>F</math>. If <math>D</math> denotes the total fluid domain, then potential <math>\Phi</math> satisfies

<center><math>\begin{matrix}

\nabla^{2}\Phi&=0 \quad\textrm{in } D, \

\frac{\partial\Phi}{\partial n} &= 0 \quad\textrm{on } \partial D,\

\Phi&=\Psi \quad\textrm{on } F

\end{matrix}</math></center>

where <math>\partial D</math> includes the bed and the structure surface. The Dirichlet to Neumann map <math>\partial_{\mathbf{n}}\mathbf{B}</math> defined by

<center><math>(12)

\partial_{\mathbf{n}}\mathbf{B}\Psi=\frac{\partial\Phi}{\partial z}, \quad x\in F,

</math></center>

then recovers the corresponding normal derivative to <math>\Psi</math> on the free surface.

The evolution equation

<center><math>

i\frac{\partial \mathbf{U}}{\partial t} = \mathcal{A}\mathbf{U}

</math></center>

accounts for the linear time-dependent free-surface equations using the vector

<center><math>

\mathbf{U}= \begin{pmatrix}

\Psi \

-i\eta

\end{pmatrix},

</math></center>

with <math>\eta</math> denoting the time-dependent free-surface elevation, and the operator <math>\mathcal{A}</math> which is both unitary and self-adjoint given by

<center><math>

\mathcal{A}= \begin{pmatrix}

0 & g \

\partial_{\mathbf{n}}\mathbf{B} & 0 \

\end{pmatrix}

</math></center>

A self-adjoint operator will possess an entirely real spectrum and the generalised eigenfunctions <math>\mathbf{u}=(\psi,-i\zeta)^{T}</math>, which are non-trivial solutions of

<center><math>(13)

\mathcal{A}\mathbf{u}=\omega\mathbf{u},

</math></center>

are just frequency-domain scattering solutions. (On a technical note, the word `generalised' precedes eigenfunction because the energy is unbounded.) This can be shown easily by combining the scalar equations resulting from (13) so as to give the frequency-domain free-surface condition. For a given frequency <math>\omega</math>, the free-surface condition in three-dimensions is satisfied by waves from an infinite number of directions and so for each <math>\omega</math> there is an infinite set of eigenfunctions <math>\mathbf{u}_{n}=(1,\omega/g)^{T}\psi_{n}(\mathbf{x},\omega)</math>, with the <math>n^{th}</math> eigenfunction corresponding to incident waves of the form <math>J_{n}(k r) e^{in\theta}</math>. The general solution of the time-evolution equation is, from spectral theory,

<center><math>(14)

\mathbf{U}(\mathbf{x},t)=\int^{\infty}_{-\infty}\left[ \sum_{n}f_{n}(\omega)\mathbf{u}_{n}(\mathbf{x},\omega) \right]e^{-i\omega t}\, d\omega

</math></center>

where <math>f_{n}(\omega)</math> is determined by the initial conditions now expressed as <math>\mathbf{U}(\mathbf{x},0)</math>. To obtain this expression for <math>f_{n}(\omega)</math>, apply the energy inner product to (14) evaluated at <math>t=0</math>. The eigenfunctions satisfy the orthogonality condition

<center><math>(15)

<\mathbf{u}_{m}(\mathbf{x},\omega),\mathbf{u}_{n}(\mathbf{x},\omega')>_{E}=\Lambda_{m}(\omega)\delta_{mn}\delta(\omega-\omega')

</math></center>

where <math><>_{E}</math> denotes a special energy inner product (defined with a <math>\mathcal{H}</math> subscript by \citeasnoun{meylaniwwwfb2008}) and it can be shown that

<center><math>

\Lambda_{m}(\omega)=\frac{4\pi\omega^{2}}{g k}\frac{d\omega}{dk}.

</math></center>

Therefore, it is straightforward to show that

<center><math>

f_{n}(\omega)=\frac{1}{\Lambda_{n}(\omega)}<\mathbf{U}(\mathbf{x},0),\mathbf{u}_{n}(\mathbf{x},\omega)>_{E}.

</math></center>

by evaluating the energy inner product <math><\mathbf{U}(\mathbf{x},0),\mathbf{u}_{n}(\mathbf{x},\omega)>_{E}</math> using the orthogonality relation~(15) and the definition~(14). This expression can be further simplified using the definition of the energy product but details will not be provided here. Instead, it should be noted that an analytic expression for the general solution has been obtained. Thus, for a given scattering problem it is in theory possible to obtain the motion of the free-surface and the potential on the free-surface.

The singularity expansion method involves moving the path of integration in~(14) in the <math>\omega</math>-plane across the singularities in the lower half plane Im<math>\,\omega\leq0</math> using the method of contour integration. Thus, the general solution will consist of contributions from the poles, any branch cuts, the path at infinity and the remainder of the path located below the poles in the <math>\omega</math>-plane. As explained by \citeasnoun{hazardSEM}, the contributions from infinity are assumed to be identically zero and the contribution from the remainder of the path decays faster than the pole contributions. Given that the contribution from the branch cuts are thought to be significant only for very large times an approximation to the general solution can be obtained for medium and large <math>t</math> because the contributions from the poles dominate for this range of times. Therefore, the SEM requires a knowledge of the pole structure of the integrand to approximate the complete integral. Furthermore, the behaviour of the eigenfunctions in the vicinity of the poles must be known in order to compute the residue of each pole. The scattering potential will satisfy

<center><math>

(I+T(\omega))\phi^{S}=g_{I}

</math></center>

where the form of the operator <math>T(\omega)</math> depends on the solution method chosen and <math>g_{I}</math> is determined by the incident wave. Given <math>g_{I}</math>, the scattered field is

<center><math>

\phi^{S}=(I+T(\omega))^{-1}g_{I}

</math></center>

and the pole structure of <math>\phi_{S}</math> will be inherited from the operator <math>(I+T(\omega))^{-1}</math>, referred to as the resolvent.

In the case where the resolvent operator is approximated by a matrix, it can be shown that the potential has the form

<center><math>(16)

\phi_{n}(\mathbf{x},\omega)\sim\frac{\alpha_{nj}v_{j}(\mathbf{x})}{\omega-\omega_{j}} \textrm{ as } \omega\rightarrow\omega_{j}

</math></center>

in the vicinity of the pole. The poles themselves are determined by locating values of <math>\omega</math> in the lower complex plane where the resolvent is not invertible. Thus, <math>v_{j}</math> is a generalised eigenfunction of <math>A_{j0}=I+T(\omega_{j})</math>. The integral in equation~(14) can be approximated by closing the integration path in the lower half plane and then moving the integration path across a finite number of <math>P</math> poles and summing over the contributions of these poles. As described by~\citeasnoun{hazardSEM}, any branch cuts in the complex <math>\omega</math>-plane will only be significant at very large times and the integrals at infinity are expected to be zero. Therefore, the contribution of the <math>P</math> poles closest to the real-<math>\omega</math> axis (and hence with the smallest decay rates <math>e^{-Im(\omega_{j})}</math>) will dominate in the medium term. So, after neglecting other contributions to the integral and using~(16) to determine the residues of the poles, the potential on the free-surface will be given by

<center><math>(17)

\Psi(\mathbf{x},t)\approx -2\pi\sum_{j}\left[ \sum_{n}f_{n}(\omega_{j})\alpha_{nj}\right]v_{j}(\mathbf{x},0)e^{-i\omega_{j}t}.

</math></center>

Although full details of how this expression can be evaluated are not given here, examples of applications of this method are given by~\citeasnoun{meylaniwwwfb2007}, \citeasnoun{meylaniwwwfb2008} and \citeasnoun{meylan2002}. Each of these papers also contain more detailed instructions regarding the computational aspects of the method, such as determining the locations of the poles.

\citeasnoun{hazardSEM} notes that the SEM is a non-rigorous method and requires some heuristic arguments to justify its use. However, in the specific cases considered the results are generally accurate for a large range of times and much frequency-domain information regarding resonances is inherent in the time-domain solution. At present, the method only applies to scattering problems; however, it is hoped that it can be extended to radiation and coupled motion problems. It is not clear how the forces on the structure will be determined.

\f1

}

{\rtf1\ansi\ansicpg1252\deff0\deflang2057{\fonttbl{\f0\fnil\fcharset0 Courier New;}{\f1\fswiss\fcharset0 Arial;}}

{\*\generator Msftedit 5.41.21.2508;}\viewkind4\uc1d\f0\fs20=Review of time-domain models=

==Frequency domain definitions==(1)

Linearised water-wave problems were first investigated in the frequency domain wherein all motions are assumed to be time-harmonic and to have begun from <math>t= -\infty</math> . The time-dependence can be removed in a straightforward manner by substituting for the potential and any structural motions using

<center><math>(2)

\begin{split}

\Phi(\mbf{x},t)=Re \{\phi(\mbf{x},\omega) e^{-i\omega t}\} \

V(\mbf{x},t)=Re \{ v(\mbf{x},\omega) e^{-i\omega t}\}

\end{split}

</math></center>

and this results in a considerable simplification of the problem. The complex amplitudes of the dynamic quantities vary with frequency and by removing the time-dependence the governing equations and boundary conditions will feature these terms only.
However, \citeasnoun{mciver2003} have shown that the frequency-domain potential can also be defined using the following operations on the time-domain solution:

*take a Laplace transform of the time-domain potential
<center><math>
\hat{\phi}(\mbf{x},s)=\int^{\infty}_{0}\Phi(\mbf{x},t)e^{-s t}\, dt ,\quad \textrm{Re } s >0;
</math></center>

*apply the change of variables <math>s=-i\omega</math> so that
<center><math>
\phi(\mbf{x},\omega)=\hat{\phi}(\mbf{x},-i\omega).
</math></center>

*noting that <math>\phi(\mbf{x},-\omega)=\bar{\phi}(\mbf{x},\omega)</math>, the inverse Fourier transform is given by
<center><math>(3)
\Phi(\mbf{x},t)=\frac{1}{2\pi}\int^{\infty}_{-\infty}\phi(\mbf{x},\omega)e^{-i\omega t} d\omega =\frac{1}{\pi}Re\int^{\infty}_{0}\phi(\mbf{x},\omega)e^{-i\omega t}\,d\omega
</math></center>
where the path of integration must pass over any singularities of <math>\phi</math> that lie on the real axis and it has been assumed that there is no motion prior to <math>t=0</math>, i.e. <math>\Phi(\mbf{x},t)=0</math> for <math>t<0</math>.

If <math>\omega</math> is real then <math>\phi(\mbf{x},\omega)</math> is just the frequency domain potential. By the principle of causality, which precludes the existence of the response before the cause so that <math>\Phi(\mbf{x},t)=0</math> for <math>t<0</math> , no singularities can exist in <math>\textrm{Im } \omega>0</math>. Therefore, if it is assumed that no singularities exist on the real axis the inverse transform will integrate the frequency-domain potential over the full range of <math>\omega</math>. With this approach, the frequency-domain equations are obtained by applying the Fourier transform to the time-dependent equations. However, unlike the equations resulting from the assumptions~(2) of time-harmonic motion from <math>t= -\infty</math>, the initial conditions will be present in the frequency-domain equations arising from the Fourier transformation. Although, the latter equations are more mathematically consistent it is usual to set all initial conditions to zero before attempting to solve the equations even if this is not the case. Nevertheless, in the frequency-domain analysis of resonances in coupled motion problems \citeasnoun{pmciver2005} retains the initial condition terms in the equation of motion for the structure and obtains some important results regarding the resonant behaviour of the motion. Therefore, the retention of the initial condition terms depends on the application of the subsequent equations.

Many analytical and approximate solutions exist for frequency-domain problems involving structures with simple geometries. Thus, in the simplest frequency-domain radiation and diffraction problems it is possible to completely determine the velocity potential analytically. The hydrodynamic forces, which are crucial in the context of marine engineering, can then be calculated directly from the potential. For more general geometries this is not possible and the hydrodynamic forces must be computed using numerical methods. Nevertheless, it is useful to consider the analytical expressions for the various potentials and forces in the frequency-domain problem.

In a linearised coupled motion problem, it is customary to decompose the total velocity potential into a scattering potential <math>\phi^{S}</math> and a radiation potential <math>\phi^{R}</math>. Thus, the coupled motion problem will require the solution of the radiation problem and the scattering problem. The radiation potential can be further decomposed into a sum over the six modes of motion so that the total frequency domain potential is

<center><math>(4)

\phi=\phi^{S}+\sum_{\mu}v_{\mu}\phi_{\mu}

</math></center>

where <math>u_{\mu}</math> is the complex amplitude of the generalised velocity in the <math>\mu</math> direction and <math>\phi_{\mu}</math> describes the fluid response due to the forced oscillations in mode <math>\mu</math> with unit velocity amplitude. The velocities in each mode will be determined from the frequency domain equation of motion for the body. However, this requires the computation of the radiation and exciting forces on the structure. Therefore, it is necessary to consider the radiation problem prior to the coupled motion equation. It is first important to note that, for a given mode <math>\mu</math> the boundary condition on the structure will be

<center><math>

\frac{tial\phi_{\mu}}{tial n}=n_{\mu},

</math></center>

because the total velocity is

<center><math>

v(\omega)=\sum_{\mu}v_{\mu}n_{\mu}

</math></center>

where <math>n_{\mu}</math> is the <math>\mu</math> component of the generalised normal and <math>v_{\mu}</math> is the component of the generalised complex velocity amplitude in this direction. The hydrodynamic force on the structure in the <math>\mu</math> direction due to the fluid response to the forced oscillations is

<center><math>(5)

F_{\mu}^{R}=i\omega\rho\iint_{S_{B}}\phi^{R}n_{\mu} \,dS

</math></center>

and the force due to the diffraction of an incident wave by the fixed structure is

<center><math>(6)

F_{\mu}^{S}=i\omega\rho\iint_{S_{B}}\phi^{S}n_{\mu} \,dS

</math></center>

where the time-dependence has been removed in both cases. By expressing the radiation potential as a sum over the individual modes, the radiation force can written as <math>\sum_{\nu} v_{\nu}f_{\nu\mu}</math> where

<center><math>

f_{\nu\mu}=i\omega\rho\iint_{\Gamma} \phi_{\nu}n_{\mu} \,dS.

</math></center>

It is conventional to decompose the radiation force into a term featuring the added mass matrix <math>a_{\alpha\beta}</math> and a term featuring the damping matrix <math>b_{\alpha\beta}</math> as follows

<center><math>(7)

f_{\beta\alpha}=i\omega(a_{\beta\alpha}+\frac{ib_{\beta\alpha}}{\omega})

</math></center>

where the added mass and damping coefficients are in phase with the acceleration and velocity respectively. These real quantities depend only on the geometry of the structure and the oscillation frequency <math>\omega</math> and describe many of the properties of the structure. In particular, (as will be shown see later) if the variation of these coefficients with frequency is known then we can predict the resonant behaviour of the structure in the time-domain as well as the frequency domain. The computation of these quantities was considered to be very important in marine engineering and naval architecture and so a variety of different computer codes were developed to compute the coefficients efficiently. The boundary element method forms the basis of most of the computer code algorithms, the most widely-used example being WAMIT (see http:<math>\backslash\backslash</math>www.wamit.com) which determines frequency domain solutions for prescribed structures and structure motions.

==Frequency-domain equation of motion==(8)

The radiation force coefficients also play a significant role in determining the motion of a floating structure possibly subject to mooring and applied forces. To obtain the correct equation of motion in the frequency domain it is important to adopt the approach similar to that taken by~\citeasnoun{pmciver2005}, i.e. keep the initial condition terms in the Fourier transform operations. Therefore, if the Fourier transform of the velocity is <math>v_{\mu}(\omega)</math> then the Fourier transform of the acceleration is given by

<center><math>

\int_{0}^{\infty}\ddot{X}_{\mu}(t)e^{i\omega t}dt=-i\omega v_{\mu}(\omega) - V_{\mu}(0)

</math></center>

the time-derivative of the potential obeys a similar relation

<center><math>

\int_{0}^{\infty}\frac{tial\Phi}{tial t}e^{i\omega t}dt=-i\omega \phi_{\mu}(\omega) - \Phi(\mbf{x},0).

</math></center>

Therefore, the initial conditions of the structure <math>(X_{\mu}(0),V_{\mu}(0))</math> and of the potential <math>\Phi(\mbf{x},0)</math> will be present in the Fourier transform of the equation motion which will be referred to as the frequency-domain equation of motion. If the frequency domain potential is decomposed in the same manner as before~(4) then the Fourier transform of the equation of motion~(\ref{linearisedmotion}) for the structure is

<center><math>(9)

\begin{split}

M\left[-V_{\mu}(0)-i\omega v_{\mu}(\omega)\right]&

=-\rho\iint_{\Gamma}(-\Phi(\mbf{x},0))n_{\mu}dS+F^{S}_{\mu}(\omega)+\

i\omega \sum_{\nu}&(f_{\mu\nu}(\omega)+i\gamma_{\mu\nu}/\omega)v_{\nu}(\omega)-

\sum_{\nu} c_{\mu\nu}\frac{v(\omega)+X(0)}{-i\omega}+f^{A}_{\mu}(\omega)

\end{split}

</math></center>

for <math>\mu=1,\ldots,6</math>, where <math>f^{A}_{\mu}(\omega)</math> is the Fourier transform of the applied force <math>F_{\mu}(t)</math> in equation~(\ref{linearisedmotion}). Although it is assumed that <math>\Phi(\mbf{x},t)=0</math> for <math>t<0</math>, for a non-zero initial velocity <math>\lim_{t^{+}\rightarrow 0}\Phi(\mbf{x},t)\neq 0</math> because of the impulsive pressure on the free-surface resulting from the initial motion of the pressure. Therefore, the first term on the right-hand side of the above equation is not identically zero, rather is given by

<center><math>

\rho\iint_{\Gamma}(-\Phi(\mbf{x},0))n_{\mu}dS=a(\infty)V(0)

</math></center>

(see \citeasnoun{ccmei2}, \S~8.12.1)

where <math>a(\infty)=\lim_{\omega\rightarrow\infty}a(\omega)</math> is the infinite frequency added mass.

The frequency-domain equation is usually re-expressed in the following form

<center><math>

\begin{split}

\sum_{\nu}&\{c_{\mu\nu}-\omega^{2}\left[M_{\mu\mu}\delta_{\mu\nu}+a_{\mu\nu}(\omega)+i(b_{\mu\nu}+\gamma_{\mu\nu})/\omega\right]\}v_{\nu}(\omega) \

=& -i\omega\left[ F_{\mu}^{S}(\omega)+f_{\mu}(\omega)+(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)\right]-\sum_{\nu}c_{\mu\nu}X_{\nu}(0)\end{split}

</math></center>

for <math>\mu=1,\ldots,6</math>. If it is assumed that the structure can only move in one mode of motion and that there is no incident wave or applied force then the complex velocity will be given by

<center><math>(10)

v_{\mu}(\omega)=\frac{-i\omega(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)-c_{\mu\mu}X_{\mu}(0)}{c_{\mu\mu}-\omega^{2}\left[M_{\mu\mu}+a_{\mu\mu}(\omega)+i(b_{\mu\mu}+\gamma_{\mu\mu})/\omega\right]}.

</math></center>

As described by \citeasnoun{mciver2006}, the zero of the denominator gives the location of the pole referred to as the motion resonance. The complex force coefficient <math>f_{\mu\nu}(\omega)</math> will also possess a pole, however this resonance is annulled by the frequency-dependence of the complex velocity (see \S~6 of \citeasnoun{mciver2006}) and so the motion, in the absence of incident waves, is dominated by the motion resonance term. Solving equation~(10) for <math>v(\omega)</math> then the time-domain velocity can be recovered using the inverse transform

<center><math>

V(t)=\frac{1}{\pi} Re \int^{\infty}_{0}v(\omega)e^{-i\omega t} dt.

</math></center>

==Integro-differential equation method==

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by <math>\delta(t-\tau)V_{\alpha}(\tau)</math> where <math>\dot{X_{\alpha}}=V_{\alpha}</math>. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function <math>V_{\alpha}(t)</math> an integral of the impulse response over the range <math>-\infty<\tau<\infty</math> must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

<center><math>(11)

\mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau

</math></center>

where <math>m_{\beta\alpha}(\infty)</math> is an infinite frequency added mass coefficient and <math>L_{\beta\alpha}(t-\tau)</math> is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. <math>V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \}</math>, the impulse response function <math>L_{\beta\alpha}</math> can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

<center><math>

\begin{split}

a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, d\tau \

b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, d\tau

\end{split}

</math></center>

and so if the added mass or damping coefficients are known for all frequencies <math>0\leq\omega\leq\infty</math>, the impulse response or memory function <math>L_{\alpha\beta}(t)</math> can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

<center><math>

\left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t),

</math></center>
for <math>\alpha=1,\ldots,6</math>, where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency <math>\omega</math> on the interval <math>[0,\infty)</math>. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

==Generalised eigenfunction expansion method==

The singularity expansion method (SEM) is based on a general scattering theory for general wave problems and has been adapted by \citeasnoun{hazardSEM} and \citeasnoun{meylaniwwwfb2008}

to problems in hydrodynamics and hydro-elasticity. The central concept in the theory is to express the governing equations for the scattering problem in terms of an evolution equation involving a unitary operator and to use the generalised eigenfunctions and eigenfrequencies of this operator to give an approximation to the time-domain solution of the scattering problem. To give a brief overview of the method the scattering equations must be re-expressed using a harmonic lifting operator <math>\mbf{B}</math> which maps a function defined on the free-surface onto the total fluid domain. This harmonic lifting operator is denoted by <math>\mbf{B}</math> so that <math>\mbf{B}\Psi=\Phi</math> where <math>\Psi=\Phi</math> on the free-surface <math>F</math>. If <math>D</math> denotes the total fluid domain, then potential <math>\Phi</math> satisfies

<center><math>\begin{matrix}

\nabla^{2}\Phi&=0 \quad\textrm{in } D, \

\frac{tial\Phi}{tial n} &= 0 \quad\textrm{on } tial D,\

\Phi&=\Psi \quad\textrm{on } F

\end{matrix}</math></center>

where <math>tial D</math> includes the bed and the structure surface. The Dirichlet to Neumann map <math>tial_{\mbf{n}}\mbf{B}</math> defined by

<center><math>(12)

tial_{\mbf{n}}\mbf{B}\Psi=\frac{tial\Phi}{tial z}, \quad x\in F,

</math></center>

then recovers the corresponding normal derivative to <math>\Psi</math> on the free surface.

The evolution equation

<center><math>

i\frac{tial \mbf{U}}{tial t} = \mathcal{A}\mbf{U}

</math></center>

accounts for the linear time-dependent free-surface equations using the vector

<center><math>

\mbf{U}= \begin{pmatrix}

\tab \Psi \

\tab -i\eta

\end{pmatrix},

</math></center>

with <math>\eta</math> denoting the time-dependent free-surface elevation, and the operator <math>\mathcal{A}</math> which is both unitary and self-adjoint given by

<center><math>

[[Category:Linear Water-Wave Theory]]

{{complete page}}

Category:Time-Dependent Linear Water Waves

2009-08-20T23:05:49Z

Cfitzgerald: /* Time domain problem for a floating structure */

Generally the focus of research is on the [[Frequency Domain Problem]]. The time-domain
problem can be solved by [[Generalised Eigenfunction Expansion]] or by
an [[Expansion over the Resonances]] or using [[Memory Effect Function]].

== Introduction ==

Analytical solutions of the linearised time-domain equations for a coupled-motion or forced-motion problems are very rare. In fact, only a small number of analytic time-domain solutions have been obtained (see the papers by [[Kennard 1949]] and [[McIver 1994]]) for structures with simple geometries. Therefore, a variety of numerical methods have been developed to solve initial-value time-domain wave-structure interaction problems. The Fourier transform enables the time and frequency domain quantities to be related and the frequency domain forces and potentials have important roles in the development of the numerical time-domain solution methods. This can be partly attributed to the fact that, in the past, analytical or semi-analytical solutions to linearised water-wave problems were more easily obtained in the frequency domain given that the assumption of time-harmonic motion results in a significant simplification of the interaction equations. Therefore, it was natural to develop time-domain solutions based on frequency domain results and the Fourier transform relation between the domains.

== Time domain problem for a floating structure ==
To begin with, it is useful to introduce the general time-dependent water wave problem. Therefore, the the time-domain equations for the motion of a floating structure must be quoted. From these equations, the forced motion and fixed body interaction problems (i.e. the radiation and scattering problems) can be derived. Furthermore, the corresponding frequency-domain equations can be derived from the time-domain in order to illustrate the important relation between the two. For a semi-submerged, vertically axisymmetric structure the time-domain linearised equations are
<center><math>
\Delta\Phi\left( \mathbf{x,}t\right) =0,\ \ \mathbf{x}\in\Omega,
</math></center>
<center><math>
\partial_{n}\Phi=0,\ \ z=-h,
</math></center>
<center><math>
\partial_{n}\Phi=\sum\dot{X}_{\nu}n_{\nu},\ \ \text{on } \Gamma,
</math></center>
where <math>\Phi</math> is the velocity potential for the fluid, <math>\Gamma</math> is the still water position of the wetted surface of the structure. At the free surface <math>F</math>, the potential must obey the kinematic condition
<center><math>
\partial_{t}\zeta=\partial_{n}\Phi,\ \ z=0,\ x\in F,
</math></center>
and the dynamic condition (the linearized Bernoulli equation)
<center><math>
\zeta = -(1/g)\partial_{t}\Phi,\ \ z=0,\ x\in F,
</math></center>
where <math>\zeta</math> is the free-surface elevation. Initial conditions for <math>\Phi</math> and <math>\partial_{t}\Phi</math> must also be specified. These fluid motion equations must be combined with the equation of motion of the body to give the coupled motion problem involving the structure motions, described by the displacement <math>X_{\nu}</math> in mode <math> \nu</math>, and the velocity potential <math>\Phi</math>. The equation of motion for a structure moored by an arrangement of linear springs and dampers is
<center><math>
M_{\mu\mu}\ddot{X}_{\mu}=-\rho\iint_{\Gamma} \frac{\partial\Phi}{\partial t} n_{\mu}\, dS-\sum^{6}_{\nu=1}\left[ c_{\mu\nu}X_{\nu}(t)+\gamma_{\mu\nu}\dot{X}_{\nu}\right]+F_{\mu}(t), \qquad \mu=1,\ldots,6
</math></center>
where <math>M_{\mu\mu}</math> are the diagonal elements of mass matrix for the structure, i.e. <math>M_{\mu\mu}=M</math> for <math>\mu=1,2,3</math> and <math>M_{\mu\mu}=I_{\mu-3,\mu-3}</math> for <math>\mu=4,5,6</math>.The characteristics of the mooring springs and dampers are described the matrices <math>\kappa_{\mu\nu}</math> and <math>\gamma_{\mu\nu}</math> , with the spring term included in the term <math>c_{\mu\nu}=\rho g b_{\mu}\delta_{\mu\nu}+k_{\mu\nu}</math> which also describes the effect of buoyancy. The initial generalised displacements <math>X_{\mu}</math> and velocities <math>\dot{X}_{\mu}</math> of the body must be specified for all modes <math>\mu=1,\ldots,6</math> in order to solve the equation.
It is non-zero for the heave, roll and pitch modes only and these terms are <math>b_{3}=W</math>, <math>b_{4}=I^{A}_{22}+I^{V}_{3}</math> and <math>b_{5}=I^{A}_{11}+I^{V}_{3}</math> respectively, where <math>W</math> is the waterplane area and <math>I^{A}</math> is the moment of the waterplane are (see Chapter 7, Mei (1983)) and <math>I^{V}_{3}</math> is <math>z</math>-component centre of buoyancy of the structure. Thus, the coupled equations of motion for a floating structure subjected to some constraint forces have been derived. (N.B. it is assumed that the centre of rotation and the centre of mass of the structure coincide for this equation, i.e. it is assumed that the body is semi-submerged. Furthermore, any wave incidence is assumed to be along the <math>x</math>-axis.) The scattering and radiation problems are simpler than the coupled problem because the motion of the the structure is then prescribed.

{{Equations for fixed bodies in the time domain}}

==Frequency domain definitions==

Linearised water-wave problems were first investigated in the frequency domain wherein all motions are assumed to be time-harmonic and to have begun from <math>t= -\infty</math> . The time-dependence can be removed in a straightforward manner by substituting for the potential and any structural motions using
<center><math>
\Phi(\mathbf{x},t)=Re \{\phi(\mathbf{x},\omega) e^{-i\omega t}\}
</math></center>
<center><math>
V(\mathbf{x},t)=Re \{ v(\mathbf{x},\omega) e^{-i\omega t}\}
</math></center>
and this results in a considerable simplification of the problem. The complex amplitudes of the dynamic quantities vary with frequency and by removing the time-dependence the governing equations and boundary conditions will feature these terms only.
However, \citeasnoun{mciver2003} have shown that the frequency-domain potential can also be defined using the following operations on the time-domain solution:

*take a Laplace transform of the time-domain potential
<center><math>
\hat{\phi}(\mathbf{x},s)=\int^{\infty}_{0}\Phi(\mathbf{x},t)e^{-s t}\, dt ,\quad \textrm{Re } s >0;
</math></center>

*apply the change of variables <math>s=-i\omega</math> so that
<center><math>
\phi(\mathbf{x},\omega)=\hat{\phi}(\mathbf{x},-i\omega).
</math></center>

*noting that <math>\phi(\mathbf{x},-\omega)=\bar{\phi}(\mathbf{x},\omega)</math>, the inverse Fourier transform is given by
<center><math>(3)
\Phi(\mathbf{x},t)=\frac{1}{2\pi}\int^{\infty}_{-\infty}\phi(\mathbf{x},\omega)e^{-i\omega t} d\omega =\frac{1}{\pi}Re\int^{\infty}_{0}\phi(\mathbf{x},\omega)e^{-i\omega t}\,d\omega
</math></center>
where the path of integration must pass over any singularities of <math>\phi</math> that lie on the real axis and it has been assumed that there is no motion prior to <math>t=0</math>, i.e. <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math>.

If <math>\omega</math> is real then <math>\phi(\mathbf{x},\omega)</math> is just the frequency domain potential. By the principle of causality, which precludes the existence of the response before the cause so that <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math> , no singularities can exist in <math>\textrm{Im } \omega>0</math>. Therefore, if it is assumed that no singularities exist on the real axis the inverse transform will integrate the frequency-domain potential over the full range of <math>\omega</math>. With this approach, the frequency-domain equations are obtained by applying the Fourier transform to the time-dependent equations. However, unlike the equations resulting from the assumptions~(2) of time-harmonic motion from <math>t= -\infty</math>, the initial conditions will be present in the frequency-domain equations arising from the Fourier transformation. Although, the latter equations are more mathematically consistent it is usual to set all initial conditions to zero before attempting to solve the equations even if this is not the case. Nevertheless, in the frequency-domain analysis of resonances in coupled motion problems \citeasnoun{pmciver2005} retains the initial condition terms in the equation of motion for the structure and obtains some important results regarding the resonant behaviour of the motion. Therefore, the retention of the initial condition terms depends on the application of the subsequent equations.

Many analytical and approximate solutions exist for frequency-domain problems involving structures with simple geometries. Thus, in the simplest frequency-domain radiation and diffraction problems it is possible to completely determine the velocity potential analytically. The hydrodynamic forces, which are crucial in the context of marine engineering, can then be calculated directly from the potential. For more general geometries this is not possible and the hydrodynamic forces must be computed using numerical methods. Nevertheless, it is useful to consider the analytical expressions for the various potentials and forces in the frequency-domain problem.

In a linearised coupled motion problem, it is customary to decompose the total velocity potential into a scattering potential <math>\phi^{S}</math> and a radiation potential <math>\phi^{R}</math>. Thus, the coupled motion problem will require the solution of the radiation problem and the scattering problem. The radiation potential can be further decomposed into a sum over the six modes of motion so that the total frequency domain potential is
<center><math>
\phi=\phi^{S}+\sum_{\mu}v_{\mu}\phi_{\mu}
</math></center>

where <math>u_{\mu}</math> is the complex amplitude of the generalised velocity in the <math>\mu</math> direction and <math>\phi_{\mu}</math> describes the fluid response due to the forced oscillations in mode <math>\mu</math> with unit velocity amplitude. The velocities in each mode will be determined from the frequency domain equation of motion for the body. However, this requires the computation of the radiation and exciting forces on the structure. Therefore, it is necessary to consider the radiation problem prior to the coupled motion equation. It is first important to note that, for a given mode <math>\mu</math> the boundary condition on the structure will be

<center><math>

\frac{\partial\phi_{\mu}}{\partial n}=n_{\mu},

</math></center>

because the total velocity is

<center><math>

v(\omega)=\sum_{\mu}v_{\mu}n_{\mu}

</math></center>

where <math>n_{\mu}</math> is the <math>\mu</math> component of the generalised normal and <math>v_{\mu}</math> is the component of the generalised complex velocity amplitude in this direction. The hydrodynamic force on the structure in the <math>\mu</math> direction due to the fluid response to the forced oscillations is

<center><math>(5)

F_{\mu}^{R}=i\omega\rho\iint_{S_{B}}\phi^{R}n_{\mu} \,dS

</math></center>

and the force due to the diffraction of an incident wave by the fixed structure is

<center><math>(6)

F_{\mu}^{S}=i\omega\rho\iint_{S_{B}}\phi^{S}n_{\mu} \,dS

</math></center>

where the time-dependence has been removed in both cases. By expressing the radiation potential as a sum over the individual modes, the radiation force can written as <math>\sum_{\nu} v_{\nu}f_{\nu\mu}</math> where

<center><math>

f_{\nu\mu}=i\omega\rho\iint_{\Gamma} \phi_{\nu}n_{\mu} \,dS.

</math></center>

It is conventional to decompose the radiation force into a term featuring the added mass matrix <math>a_{\alpha\beta}</math> and a term featuring the damping matrix <math>b_{\alpha\beta}</math> as follows

<center><math>(7)

f_{\beta\alpha}=i\omega(a_{\beta\alpha}+\frac{ib_{\beta\alpha}}{\omega})

</math></center>

where the added mass and damping coefficients are in phase with the acceleration and velocity respectively. These real quantities depend only on the geometry of the structure and the oscillation frequency <math>\omega</math> and describe many of the properties of the structure. In particular, (as will be shown see later) if the variation of these coefficients with frequency is known then we can predict the resonant behaviour of the structure in the time-domain as well as the frequency domain. The computation of these quantities was considered to be very important in marine engineering and naval architecture and so a variety of different computer codes were developed to compute the coefficients efficiently. The boundary element method forms the basis of most of the computer code algorithms, the most widely-used example being WAMIT (see http:<math>\backslash\backslash</math>www.wamit.com) which determines frequency domain solutions for prescribed structures and structure motions.

==Frequency-domain equation of motion==

The radiation force coefficients also play a significant role in determining the motion of a floating structure possibly subject to mooring and applied forces. To obtain the correct equation of motion in the frequency domain it is important to adopt the approach similar to that taken by [[McIver 2005]], i.e. keep the initial condition terms in the Fourier transform operations. Therefore, if the Fourier transform of the velocity is <math>v_{\mu}(\omega)</math> then the Fourier transform of the acceleration is given by

<center><math>

\int_{0}^{\infty}\ddot{X}_{\mu}(t)e^{i\omega t}dt=-i\omega v_{\mu}(\omega) - V_{\mu}(0)

</math></center>

the time-derivative of the potential obeys a similar relation

<center><math>

\int_{0}^{\infty}\frac{\partial\Phi}{\partial t}e^{i\omega t}dt=-i\omega \phi_{\mu}(\omega) - \Phi(\mathbf{x},0).

</math></center>

Therefore, the ini\partial conditions of the structure <math>(X_{\mu}(0),V_{\mu}(0))</math> and of the potential <math>\Phi(\mathbf{x},0)</math> will be present in the Fourier transform of the equation motion which will be referred to as the frequency-domain equation of motion. If the frequency domain potential is decomposed in the same manner as before then the Fourier transform of the equation of motion for the structure is
<center><math>
M\left[-V_{\mu}(0)-i\omega v_{\mu}(\omega)\right]
=-\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS+F^{S}_{\mu}(\omega)+
i\omega \sum_{\nu} (f_{\mu\nu}(\omega)+i\gamma_{\mu\nu}/\omega)v_{\nu}(\omega)-
\sum_{\nu} c_{\mu\nu}\frac{v(\omega)+X(0)}{-i\omega}+f^{A}_{\mu}(\omega)
</math></center>
for <math>\mu=1,\ldots,6</math>, where <math>f^{A}_{\mu}(\omega)</math> is the Fourier transform of the applied force <math>F_{\mu}(t)</math> in equation~(\ref{linearisedmotion}). Although it is assumed that <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math>, for a non-zero initial velocity <math>\lim_{t^{+}\rightarrow 0}\Phi(\mathbf{x},t)\neq 0</math> because of the impulsive pressure on the free-surface resulting from the initial motion of the pressure. Therefore, the first term on the right-hand side of the above equation is not identically zero, rather is given by
<center><math>
\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS=a(\infty)V(0)
</math></center>
[[Mei 1983]]
where <math>a(\infty)=\lim_{\omega\rightarrow\infty}a(\omega)</math> is the infinite frequency added mass.

The frequency-domain equation is usually re-expressed in the following form
<center><math>
\sum_{\nu} \{c_{\mu\nu}-\omega^{2}\left[M_{\mu\mu}\delta_{\mu\nu}+a_{\mu\nu}(\omega)+i(b_{\mu\nu}+\gamma_{\mu\nu})/\omega\right]\}v_{\nu}(\omega)
= -i\omega\left[ F_{\mu}^{S}(\omega)+f_{\mu}(\omega)+(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)\right]-\sum_{\nu}c_{\mu\nu}X_{\nu}(0)
</math></center>
for <math>\mu=1,\ldots,6</math>. If it is assumed that the structure can only move in one mode of motion and that there is no incident wave or applied force then the complex velocity will be given by
<center><math>
v_{\mu}(\omega)=\frac{-i\omega(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)-c_{\mu\mu}X_{\mu}(0)}{c_{\mu\mu}-\omega^{2}\left[M_{\mu\mu}+a_{\mu\mu}(\omega)+i(b_{\mu\mu}+\gamma_{\mu\mu})/\omega\right]}.
</math></center>

As described by [[McIver 2006]], the zero of the denominator gives the location of the pole referred to as the motion resonance. The complex force coefficient <math>f_{\mu\nu}(\omega)</math> will also possess a pole, however this resonance is annulled by the frequency-dependence of the complex velocity (see [[McIver 2006]]) and so the motion, in the absence of incident waves, is dominated by the motion resonance term. Solving for <math>v(\omega)</math> then the time-domain velocity can be recovered using the inverse transform
<center><math>
V(t)=\frac{1}{\pi} Re \int^{\infty}_{0}v(\omega)e^{-i\omega t} dt.
</math></center>

==Integro-differential equation method==

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by <math>\delta(t-\tau)V_{\alpha}(\tau)</math> where <math>\dot{X_{\alpha}}=V_{\alpha}</math>. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function <math>V_{\alpha}(t)</math> an integral of the impulse response over the range <math>-\infty<\tau<\infty</math> must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

<center><math>(11)

\mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau

</math></center>

where <math>m_{\beta\alpha}(\infty)</math> is an infinite frequency added mass coefficient and <math>L_{\beta\alpha}(t-\tau)</math> is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. <math>V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \}</math>, the impulse response function <math>L_{\beta\alpha}</math> can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

<center><math>

\begin{split}

a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, d\tau \

b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, d\tau

\end{split}

</math></center>

and so if the added mass or damping coefficients are known for all frequencies <math>0\leq\omega\leq\infty</math>, the impulse response or memory function <math>L_{\alpha\beta}(t)</math> can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

<center><math>

\left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t),

</math></center>
for <math>\alpha=1,\ldots,6</math>, where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency <math>\omega</math> on the interval <math>[0,\infty)</math>. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

==Generalised eigenfunction expansion method==

The singularity expansion method (SEM) is based on a general scattering theory for general wave problems and has been adapted by \citeasnoun{hazardSEM} and \citeasnoun{meylaniwwwfb2008}

to problems in hydrodynamics and hydro-elasticity. The central concept in the theory is to express the governing equations for the scattering problem in terms of an evolution equation involving a unitary operator and to use the generalised eigenfunctions and eigenfrequencies of this operator to give an approximation to the time-domain solution of the scattering problem. To give a brief overview of the method the scattering equations must be re-expressed using a harmonic lifting operator <math>\mathbf{B}</math> which maps a function defined on the free-surface onto the total fluid domain. This harmonic lifting operator is denoted by <math>\mathbf{B}</math> so that <math>\mathbf{B}\Psi=\Phi</math> where <math>\Psi=\Phi</math> on the free-surface <math>F</math>. If <math>D</math> denotes the total fluid domain, then potential <math>\Phi</math> satisfies

<center><math>\begin{matrix}

\nabla^{2}\Phi&=0 \quad\textrm{in } D, \

\frac{\partial\Phi}{\partial n} &= 0 \quad\textrm{on } \partial D,\

\Phi&=\Psi \quad\textrm{on } F

\end{matrix}</math></center>

where <math>\partial D</math> includes the bed and the structure surface. The Dirichlet to Neumann map <math>\partial_{\mathbf{n}}\mathbf{B}</math> defined by

<center><math>(12)

\partial_{\mathbf{n}}\mathbf{B}\Psi=\frac{\partial\Phi}{\partial z}, \quad x\in F,

</math></center>

then recovers the corresponding normal derivative to <math>\Psi</math> on the free surface.

The evolution equation

<center><math>

i\frac{\partial \mathbf{U}}{\partial t} = \mathcal{A}\mathbf{U}

</math></center>

accounts for the linear time-dependent free-surface equations using the vector

<center><math>

\mathbf{U}= \begin{pmatrix}

\Psi \

-i\eta

\end{pmatrix},

</math></center>

with <math>\eta</math> denoting the time-dependent free-surface elevation, and the operator <math>\mathcal{A}</math> which is both unitary and self-adjoint given by

<center><math>

\mathcal{A}= \begin{pmatrix}

0 & g \

\partial_{\mathbf{n}}\mathbf{B} & 0 \

\end{pmatrix}

</math></center>

A self-adjoint operator will possess an entirely real spectrum and the generalised eigenfunctions <math>\mathbf{u}=(\psi,-i\zeta)^{T}</math>, which are non-trivial solutions of

<center><math>(13)

\mathcal{A}\mathbf{u}=\omega\mathbf{u},

</math></center>

are just frequency-domain scattering solutions. (On a technical note, the word `generalised' precedes eigenfunction because the energy is unbounded.) This can be shown easily by combining the scalar equations resulting from (13) so as to give the frequency-domain free-surface condition. For a given frequency <math>\omega</math>, the free-surface condition in three-dimensions is satisfied by waves from an infinite number of directions and so for each <math>\omega</math> there is an infinite set of eigenfunctions <math>\mathbf{u}_{n}=(1,\omega/g)^{T}\psi_{n}(\mathbf{x},\omega)</math>, with the <math>n^{th}</math> eigenfunction corresponding to incident waves of the form <math>J_{n}(k r) e^{in\theta}</math>. The general solution of the time-evolution equation is, from spectral theory,

<center><math>(14)

\mathbf{U}(\mathbf{x},t)=\int^{\infty}_{-\infty}\left[ \sum_{n}f_{n}(\omega)\mathbf{u}_{n}(\mathbf{x},\omega) \right]e^{-i\omega t}\, d\omega

</math></center>

where <math>f_{n}(\omega)</math> is determined by the initial conditions now expressed as <math>\mathbf{U}(\mathbf{x},0)</math>. To obtain this expression for <math>f_{n}(\omega)</math>, apply the energy inner product to (14) evaluated at <math>t=0</math>. The eigenfunctions satisfy the orthogonality condition

<center><math>(15)

<\mathbf{u}_{m}(\mathbf{x},\omega),\mathbf{u}_{n}(\mathbf{x},\omega')>_{E}=\Lambda_{m}(\omega)\delta_{mn}\delta(\omega-\omega')

</math></center>

where <math><>_{E}</math> denotes a special energy inner product (defined with a <math>\mathcal{H}</math> subscript by \citeasnoun{meylaniwwwfb2008}) and it can be shown that

<center><math>

\Lambda_{m}(\omega)=\frac{4\pi\omega^{2}}{g k}\frac{d\omega}{dk}.

</math></center>

Therefore, it is straightforward to show that

<center><math>

f_{n}(\omega)=\frac{1}{\Lambda_{n}(\omega)}<\mathbf{U}(\mathbf{x},0),\mathbf{u}_{n}(\mathbf{x},\omega)>_{E}.

</math></center>

by evaluating the energy inner product <math><\mathbf{U}(\mathbf{x},0),\mathbf{u}_{n}(\mathbf{x},\omega)>_{E}</math> using the orthogonality relation~(15) and the definition~(14). This expression can be further simplified using the definition of the energy product but details will not be provided here. Instead, it should be noted that an analytic expression for the general solution has been obtained. Thus, for a given scattering problem it is in theory possible to obtain the motion of the free-surface and the potential on the free-surface.

The singularity expansion method involves moving the path of integration in~(14) in the <math>\omega</math>-plane across the singularities in the lower half plane Im<math>\,\omega\leq0</math> using the method of contour integration. Thus, the general solution will consist of contributions from the poles, any branch cuts, the path at infinity and the remainder of the path located below the poles in the <math>\omega</math>-plane. As explained by \citeasnoun{hazardSEM}, the contributions from infinity are assumed to be identically zero and the contribution from the remainder of the path decays faster than the pole contributions. Given that the contribution from the branch cuts are thought to be significant only for very large times an approximation to the general solution can be obtained for medium and large <math>t</math> because the contributions from the poles dominate for this range of times. Therefore, the SEM requires a knowledge of the pole structure of the integrand to approximate the complete integral. Furthermore, the behaviour of the eigenfunctions in the vicinity of the poles must be known in order to compute the residue of each pole. The scattering potential will satisfy

<center><math>

(I+T(\omega))\phi^{S}=g_{I}

</math></center>

where the form of the operator <math>T(\omega)</math> depends on the solution method chosen and <math>g_{I}</math> is determined by the incident wave. Given <math>g_{I}</math>, the scattered field is

<center><math>

\phi^{S}=(I+T(\omega))^{-1}g_{I}

</math></center>

and the pole structure of <math>\phi_{S}</math> will be inherited from the operator <math>(I+T(\omega))^{-1}</math>, referred to as the resolvent.

In the case where the resolvent operator is approximated by a matrix, it can be shown that the potential has the form

<center><math>(16)

\phi_{n}(\mathbf{x},\omega)\sim\frac{\alpha_{nj}v_{j}(\mathbf{x})}{\omega-\omega_{j}} \textrm{ as } \omega\rightarrow\omega_{j}

</math></center>

in the vicinity of the pole. The poles themselves are determined by locating values of <math>\omega</math> in the lower complex plane where the resolvent is not invertible. Thus, <math>v_{j}</math> is a generalised eigenfunction of <math>A_{j0}=I+T(\omega_{j})</math>. The integral in equation~(14) can be approximated by closing the integration path in the lower half plane and then moving the integration path across a finite number of <math>P</math> poles and summing over the contributions of these poles. As described by~\citeasnoun{hazardSEM}, any branch cuts in the complex <math>\omega</math>-plane will only be significant at very large times and the integrals at infinity are expected to be zero. Therefore, the contribution of the <math>P</math> poles closest to the real-<math>\omega</math> axis (and hence with the smallest decay rates <math>e^{-Im(\omega_{j})}</math>) will dominate in the medium term. So, after neglecting other contributions to the integral and using~(16) to determine the residues of the poles, the potential on the free-surface will be given by

<center><math>(17)

\Psi(\mathbf{x},t)\approx -2\pi\sum_{j}\left[ \sum_{n}f_{n}(\omega_{j})\alpha_{nj}\right]v_{j}(\mathbf{x},0)e^{-i\omega_{j}t}.

</math></center>

Although full details of how this expression can be evaluated are not given here, examples of applications of this method are given by~\citeasnoun{meylaniwwwfb2007}, \citeasnoun{meylaniwwwfb2008} and \citeasnoun{meylan2002}. Each of these papers also contain more detailed instructions regarding the computational aspects of the method, such as determining the locations of the poles.

\citeasnoun{hazardSEM} notes that the SEM is a non-rigorous method and requires some heuristic arguments to justify its use. However, in the specific cases considered the results are generally accurate for a large range of times and much frequency-domain information regarding resonances is inherent in the time-domain solution. At present, the method only applies to scattering problems; however, it is hoped that it can be extended to radiation and coupled motion problems. It is not clear how the forces on the structure will be determined.

\f1

}

{\rtf1\ansi\ansicpg1252\deff0\deflang2057{\fonttbl{\f0\fnil\fcharset0 Courier New;}{\f1\fswiss\fcharset0 Arial;}}

{\*\generator Msftedit 5.41.21.2508;}\viewkind4\uc1d\f0\fs20=Review of time-domain models=

==Frequency domain definitions==(1)

Linearised water-wave problems were first investigated in the frequency domain wherein all motions are assumed to be time-harmonic and to have begun from <math>t= -\infty</math> . The time-dependence can be removed in a straightforward manner by substituting for the potential and any structural motions using

<center><math>(2)

\begin{split}

\Phi(\mbf{x},t)=Re \{\phi(\mbf{x},\omega) e^{-i\omega t}\} \

V(\mbf{x},t)=Re \{ v(\mbf{x},\omega) e^{-i\omega t}\}

\end{split}

</math></center>

and this results in a considerable simplification of the problem. The complex amplitudes of the dynamic quantities vary with frequency and by removing the time-dependence the governing equations and boundary conditions will feature these terms only.
However, \citeasnoun{mciver2003} have shown that the frequency-domain potential can also be defined using the following operations on the time-domain solution:

*take a Laplace transform of the time-domain potential
<center><math>
\hat{\phi}(\mbf{x},s)=\int^{\infty}_{0}\Phi(\mbf{x},t)e^{-s t}\, dt ,\quad \textrm{Re } s >0;
</math></center>

*apply the change of variables <math>s=-i\omega</math> so that
<center><math>
\phi(\mbf{x},\omega)=\hat{\phi}(\mbf{x},-i\omega).
</math></center>

*noting that <math>\phi(\mbf{x},-\omega)=\bar{\phi}(\mbf{x},\omega)</math>, the inverse Fourier transform is given by
<center><math>(3)
\Phi(\mbf{x},t)=\frac{1}{2\pi}\int^{\infty}_{-\infty}\phi(\mbf{x},\omega)e^{-i\omega t} d\omega =\frac{1}{\pi}Re\int^{\infty}_{0}\phi(\mbf{x},\omega)e^{-i\omega t}\,d\omega
</math></center>
where the path of integration must pass over any singularities of <math>\phi</math> that lie on the real axis and it has been assumed that there is no motion prior to <math>t=0</math>, i.e. <math>\Phi(\mbf{x},t)=0</math> for <math>t<0</math>.

If <math>\omega</math> is real then <math>\phi(\mbf{x},\omega)</math> is just the frequency domain potential. By the principle of causality, which precludes the existence of the response before the cause so that <math>\Phi(\mbf{x},t)=0</math> for <math>t<0</math> , no singularities can exist in <math>\textrm{Im } \omega>0</math>. Therefore, if it is assumed that no singularities exist on the real axis the inverse transform will integrate the frequency-domain potential over the full range of <math>\omega</math>. With this approach, the frequency-domain equations are obtained by applying the Fourier transform to the time-dependent equations. However, unlike the equations resulting from the assumptions~(2) of time-harmonic motion from <math>t= -\infty</math>, the initial conditions will be present in the frequency-domain equations arising from the Fourier transformation. Although, the latter equations are more mathematically consistent it is usual to set all initial conditions to zero before attempting to solve the equations even if this is not the case. Nevertheless, in the frequency-domain analysis of resonances in coupled motion problems \citeasnoun{pmciver2005} retains the initial condition terms in the equation of motion for the structure and obtains some important results regarding the resonant behaviour of the motion. Therefore, the retention of the initial condition terms depends on the application of the subsequent equations.

Many analytical and approximate solutions exist for frequency-domain problems involving structures with simple geometries. Thus, in the simplest frequency-domain radiation and diffraction problems it is possible to completely determine the velocity potential analytically. The hydrodynamic forces, which are crucial in the context of marine engineering, can then be calculated directly from the potential. For more general geometries this is not possible and the hydrodynamic forces must be computed using numerical methods. Nevertheless, it is useful to consider the analytical expressions for the various potentials and forces in the frequency-domain problem.

In a linearised coupled motion problem, it is customary to decompose the total velocity potential into a scattering potential <math>\phi^{S}</math> and a radiation potential <math>\phi^{R}</math>. Thus, the coupled motion problem will require the solution of the radiation problem and the scattering problem. The radiation potential can be further decomposed into a sum over the six modes of motion so that the total frequency domain potential is

<center><math>(4)

\phi=\phi^{S}+\sum_{\mu}v_{\mu}\phi_{\mu}

</math></center>

where <math>u_{\mu}</math> is the complex amplitude of the generalised velocity in the <math>\mu</math> direction and <math>\phi_{\mu}</math> describes the fluid response due to the forced oscillations in mode <math>\mu</math> with unit velocity amplitude. The velocities in each mode will be determined from the frequency domain equation of motion for the body. However, this requires the computation of the radiation and exciting forces on the structure. Therefore, it is necessary to consider the radiation problem prior to the coupled motion equation. It is first important to note that, for a given mode <math>\mu</math> the boundary condition on the structure will be

<center><math>

\frac{tial\phi_{\mu}}{tial n}=n_{\mu},

</math></center>

because the total velocity is

<center><math>

v(\omega)=\sum_{\mu}v_{\mu}n_{\mu}

</math></center>

where <math>n_{\mu}</math> is the <math>\mu</math> component of the generalised normal and <math>v_{\mu}</math> is the component of the generalised complex velocity amplitude in this direction. The hydrodynamic force on the structure in the <math>\mu</math> direction due to the fluid response to the forced oscillations is

<center><math>(5)

F_{\mu}^{R}=i\omega\rho\iint_{S_{B}}\phi^{R}n_{\mu} \,dS

</math></center>

and the force due to the diffraction of an incident wave by the fixed structure is

<center><math>(6)

F_{\mu}^{S}=i\omega\rho\iint_{S_{B}}\phi^{S}n_{\mu} \,dS

</math></center>

where the time-dependence has been removed in both cases. By expressing the radiation potential as a sum over the individual modes, the radiation force can written as <math>\sum_{\nu} v_{\nu}f_{\nu\mu}</math> where

<center><math>

f_{\nu\mu}=i\omega\rho\iint_{\Gamma} \phi_{\nu}n_{\mu} \,dS.

</math></center>

It is conventional to decompose the radiation force into a term featuring the added mass matrix <math>a_{\alpha\beta}</math> and a term featuring the damping matrix <math>b_{\alpha\beta}</math> as follows

<center><math>(7)

f_{\beta\alpha}=i\omega(a_{\beta\alpha}+\frac{ib_{\beta\alpha}}{\omega})

</math></center>

where the added mass and damping coefficients are in phase with the acceleration and velocity respectively. These real quantities depend only on the geometry of the structure and the oscillation frequency <math>\omega</math> and describe many of the properties of the structure. In particular, (as will be shown see later) if the variation of these coefficients with frequency is known then we can predict the resonant behaviour of the structure in the time-domain as well as the frequency domain. The computation of these quantities was considered to be very important in marine engineering and naval architecture and so a variety of different computer codes were developed to compute the coefficients efficiently. The boundary element method forms the basis of most of the computer code algorithms, the most widely-used example being WAMIT (see http:<math>\backslash\backslash</math>www.wamit.com) which determines frequency domain solutions for prescribed structures and structure motions.

==Frequency-domain equation of motion==(8)

The radiation force coefficients also play a significant role in determining the motion of a floating structure possibly subject to mooring and applied forces. To obtain the correct equation of motion in the frequency domain it is important to adopt the approach similar to that taken by~\citeasnoun{pmciver2005}, i.e. keep the initial condition terms in the Fourier transform operations. Therefore, if the Fourier transform of the velocity is <math>v_{\mu}(\omega)</math> then the Fourier transform of the acceleration is given by

<center><math>

\int_{0}^{\infty}\ddot{X}_{\mu}(t)e^{i\omega t}dt=-i\omega v_{\mu}(\omega) - V_{\mu}(0)

</math></center>

the time-derivative of the potential obeys a similar relation

<center><math>

\int_{0}^{\infty}\frac{tial\Phi}{tial t}e^{i\omega t}dt=-i\omega \phi_{\mu}(\omega) - \Phi(\mbf{x},0).

</math></center>

Therefore, the initial conditions of the structure <math>(X_{\mu}(0),V_{\mu}(0))</math> and of the potential <math>\Phi(\mbf{x},0)</math> will be present in the Fourier transform of the equation motion which will be referred to as the frequency-domain equation of motion. If the frequency domain potential is decomposed in the same manner as before~(4) then the Fourier transform of the equation of motion~(\ref{linearisedmotion}) for the structure is

<center><math>(9)

\begin{split}

M\left[-V_{\mu}(0)-i\omega v_{\mu}(\omega)\right]&

=-\rho\iint_{\Gamma}(-\Phi(\mbf{x},0))n_{\mu}dS+F^{S}_{\mu}(\omega)+\

i\omega \sum_{\nu}&(f_{\mu\nu}(\omega)+i\gamma_{\mu\nu}/\omega)v_{\nu}(\omega)-

\sum_{\nu} c_{\mu\nu}\frac{v(\omega)+X(0)}{-i\omega}+f^{A}_{\mu}(\omega)

\end{split}

</math></center>

for <math>\mu=1,\ldots,6</math>, where <math>f^{A}_{\mu}(\omega)</math> is the Fourier transform of the applied force <math>F_{\mu}(t)</math> in equation~(\ref{linearisedmotion}). Although it is assumed that <math>\Phi(\mbf{x},t)=0</math> for <math>t<0</math>, for a non-zero initial velocity <math>\lim_{t^{+}\rightarrow 0}\Phi(\mbf{x},t)\neq 0</math> because of the impulsive pressure on the free-surface resulting from the initial motion of the pressure. Therefore, the first term on the right-hand side of the above equation is not identically zero, rather is given by

<center><math>

\rho\iint_{\Gamma}(-\Phi(\mbf{x},0))n_{\mu}dS=a(\infty)V(0)

</math></center>

(see \citeasnoun{ccmei2}, \S~8.12.1)

where <math>a(\infty)=\lim_{\omega\rightarrow\infty}a(\omega)</math> is the infinite frequency added mass.

The frequency-domain equation is usually re-expressed in the following form

<center><math>

\begin{split}

\sum_{\nu}&\{c_{\mu\nu}-\omega^{2}\left[M_{\mu\mu}\delta_{\mu\nu}+a_{\mu\nu}(\omega)+i(b_{\mu\nu}+\gamma_{\mu\nu})/\omega\right]\}v_{\nu}(\omega) \

=& -i\omega\left[ F_{\mu}^{S}(\omega)+f_{\mu}(\omega)+(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)\right]-\sum_{\nu}c_{\mu\nu}X_{\nu}(0)\end{split}

</math></center>

for <math>\mu=1,\ldots,6</math>. If it is assumed that the structure can only move in one mode of motion and that there is no incident wave or applied force then the complex velocity will be given by

<center><math>(10)

v_{\mu}(\omega)=\frac{-i\omega(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)-c_{\mu\mu}X_{\mu}(0)}{c_{\mu\mu}-\omega^{2}\left[M_{\mu\mu}+a_{\mu\mu}(\omega)+i(b_{\mu\mu}+\gamma_{\mu\mu})/\omega\right]}.

</math></center>

As described by \citeasnoun{mciver2006}, the zero of the denominator gives the location of the pole referred to as the motion resonance. The complex force coefficient <math>f_{\mu\nu}(\omega)</math> will also possess a pole, however this resonance is annulled by the frequency-dependence of the complex velocity (see \S~6 of \citeasnoun{mciver2006}) and so the motion, in the absence of incident waves, is dominated by the motion resonance term. Solving equation~(10) for <math>v(\omega)</math> then the time-domain velocity can be recovered using the inverse transform

<center><math>

V(t)=\frac{1}{\pi} Re \int^{\infty}_{0}v(\omega)e^{-i\omega t} dt.

</math></center>

==Integro-differential equation method==

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by <math>\delta(t-\tau)V_{\alpha}(\tau)</math> where <math>\dot{X_{\alpha}}=V_{\alpha}</math>. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function <math>V_{\alpha}(t)</math> an integral of the impulse response over the range <math>-\infty<\tau<\infty</math> must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

<center><math>(11)

\mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau

</math></center>

where <math>m_{\beta\alpha}(\infty)</math> is an infinite frequency added mass coefficient and <math>L_{\beta\alpha}(t-\tau)</math> is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. <math>V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \}</math>, the impulse response function <math>L_{\beta\alpha}</math> can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

<center><math>

\begin{split}

a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, d\tau \

b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, d\tau

\end{split}

</math></center>

and so if the added mass or damping coefficients are known for all frequencies <math>0\leq\omega\leq\infty</math>, the impulse response or memory function <math>L_{\alpha\beta}(t)</math> can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

<center><math>

\left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t),

</math></center>
for <math>\alpha=1,\ldots,6</math>, where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency <math>\omega</math> on the interval <math>[0,\infty)</math>. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

==Generalised eigenfunction expansion method==

The singularity expansion method (SEM) is based on a general scattering theory for general wave problems and has been adapted by \citeasnoun{hazardSEM} and \citeasnoun{meylaniwwwfb2008}

to problems in hydrodynamics and hydro-elasticity. The central concept in the theory is to express the governing equations for the scattering problem in terms of an evolution equation involving a unitary operator and to use the generalised eigenfunctions and eigenfrequencies of this operator to give an approximation to the time-domain solution of the scattering problem. To give a brief overview of the method the scattering equations must be re-expressed using a harmonic lifting operator <math>\mbf{B}</math> which maps a function defined on the free-surface onto the total fluid domain. This harmonic lifting operator is denoted by <math>\mbf{B}</math> so that <math>\mbf{B}\Psi=\Phi</math> where <math>\Psi=\Phi</math> on the free-surface <math>F</math>. If <math>D</math> denotes the total fluid domain, then potential <math>\Phi</math> satisfies

<center><math>\begin{matrix}

\nabla^{2}\Phi&=0 \quad\textrm{in } D, \

\frac{tial\Phi}{tial n} &= 0 \quad\textrm{on } tial D,\

\Phi&=\Psi \quad\textrm{on } F

\end{matrix}</math></center>

where <math>tial D</math> includes the bed and the structure surface. The Dirichlet to Neumann map <math>tial_{\mbf{n}}\mbf{B}</math> defined by

<center><math>(12)

tial_{\mbf{n}}\mbf{B}\Psi=\frac{tial\Phi}{tial z}, \quad x\in F,

</math></center>

then recovers the corresponding normal derivative to <math>\Psi</math> on the free surface.

The evolution equation

<center><math>

i\frac{tial \mbf{U}}{tial t} = \mathcal{A}\mbf{U}

</math></center>

accounts for the linear time-dependent free-surface equations using the vector

<center><math>

\mbf{U}= \begin{pmatrix}

\tab \Psi \

\tab -i\eta

\end{pmatrix},

</math></center>

with <math>\eta</math> denoting the time-dependent free-surface elevation, and the operator <math>\mathcal{A}</math> which is both unitary and self-adjoint given by

<center><math>

[[Category:Linear Water-Wave Theory]]

{{complete page}}

Category:Time-Dependent Linear Water Waves

2009-08-20T05:32:48Z

Cfitzgerald: Additional time-domain section for body motion.

Generally the focus of research is on the [[Frequency Domain Problem]]. The time-domain
problem can be solved by [[Generalised Eigenfunction Expansion]] or by
an [[Expansion over the Resonances]] or using [[Memory Effect Function]].

== Introduction ==

Analytical solutions of the linearised time-domain equations for a coupled-motion or forced-motion problems are very rare. In fact, only a small number of analytic time-domain solutions have been obtained (see the papers by [[Kennard 1949]] and [[McIver 1994]]) for structures with simple geometries. Therefore, a variety of numerical methods have been developed to solve initial-value time-domain wave-structure interaction problems. The Fourier transform enables the time and frequency domain quantities to be related and the frequency domain forces and potentials have important roles in the development of the numerical time-domain solution methods. This can be partly attributed to the fact that, in the past, analytical or semi-analytical solutions to linearised water-wave problems were more easily obtained in the frequency domain given that the assumption of time-harmonic motion results in a significant simplification of the interaction equations. Therefore, it was natural to develop time-domain solutions based on frequency domain results and the Fourier transform relation between the domains.

== Time domain problem for a floating structure ==
To begin with, it is useful to introduce the general time-dependent water wave problem. Therefore, the the time-domain equations for the motion of a floating structure must be quoted. From these equations, the forced motion and fixed body interaction problems (i.e. the radiation and scattering problems) can be derived. Furthermore, the corresponding frequency-domain equations can be derived from the time-domain in order to illustrate the important relation between the two. For a semi-submerged, vertically axisymmetric structure the time-domain linearised equations are
<center><math>
\Delta\Phi\left( \mathbf{x,}t\right) =0,\ \ \mathbf{x}\in\Omega,
</math></center>
<center><math>
\partial_{n}\Phi=0,\ \ z=-h,
</math></center>
<center><math>
\partial_{n}\Phi=\sum\dot{X}_{\nu}n_{\nu},\ \ on \Gamma,
</math></center>
where <math>\Phi</math> is the velocity potential for the fluid, <math>\Gamma</math> is the still water position of the wetted surface of the structure. At the free surface <math>F</math>, the potential must obey the kinematic condition
<center><math>
\partial_{t}\zeta=\partial_{n}\Phi,\ \ z=0,\ x\in F,
</math></center>
and the dynamic condition (the linearized Bernoulli equation)
<center><math>
\zeta = -(1/g)\partial_{t}\Phi,\ \ z=0,\ x\in F,
</math></center>
where <math>\zeta</math> is the free-surface elevation. Initial conditions for <math>\Phi</math> and <math>\partial_{t}\Phi</math> must also be specified. These fluid motion equations must be combined with the equation of motion of the body to give the coupled motion problem involving the structure motions, described by the displacement <math>X_{\nu}</math> in mode <math> \nu</math>, and the velocity potential <math>\Phi</math>. The equation of motion for a structure moored by an arrangement of linear springs and dampers is
<center><math>
M_{\mu\mu}\ddot{X}_{\mu}=-\rho\iint_{\Gamma} \frac{\partial\Phi}{\partial t} n_{\mu}\, dS-\sum^{6}_{\nu=1}\left[ c_{\mu\nu}X_{\nu}(t)+\gamma_{\mu\nu}\dot{X}_{\nu}\right]+F_{\mu}(t), \qquad \mu=1,\ldots,6
</math></center>
where <math>M_{\mu\mu}</math> are the diagonal elements of mass matrix for the structure, i.e. <math>M_{\mu\mu}=M</math> for <math>\mu=1,2,3</math> and <math>M_{\mu\mu}=I_{\mu-3,\mu-3}</math> for <math>\mu=4,5,6</math>.The characteristics of the mooring springs and dampers are described the matrices <math>\kappa_{\mu\nu}</math> and <math>\gamma_{\mu\nu}</math> , with the spring term included in the term <math>c_{\mu\nu}=\rho g b_{\mu}\delta_{\mu\nu}+k_{\mu\nu}</math> which also describes the effect of buoyancy. The initial generalised displacements <math>X_{\mu}</math> and velocities <math>\dot{X}_{\mu}</math> of the body must be specified for all modes <math>\mu=1,\ldots,6</math> in order to solve the equation.
It is non-zero for the heave, roll and pitch modes only and these terms are <math>b_{3}=W</math>, <math>b_{4}=I^{A}_{22}+I^{V}_{3}</math> and <math>b_{5}=I^{A}_{11}+I^{V}_{3}</math> respectively, where <math>W</math> is the waterplane area and <math>I^{A}</math> is the moment of the waterplane are (see Chapter 7, Mei (1983)) and <math>I^{V}_{3}</math> is centre of buoyancy of the structure. Thus, the coupled equations of motion for a floating structure subjected to some constraint forces have been derived. (N.B. it is assumed that the centre of rotation and the centre of mass of the structure coincide for this equation, i.e. it is assumed that the body is semi-submerged. Furthermore, any wave incidence is assumed to be along the <math>x</math>-axis.) The scattering and radiation problems are simpler than the coupled problem because the motion of the the structure is then prescribed.

{{Equations for fixed bodies in the time domain}}

==Frequency domain definitions==

Linearised water-wave problems were first investigated in the frequency domain wherein all motions are assumed to be time-harmonic and to have begun from <math>t= -\infty</math> . The time-dependence can be removed in a straightforward manner by substituting for the potential and any structural motions using
<center><math>
\Phi(\mathbf{x},t)=Re \{\phi(\mathbf{x},\omega) e^{-i\omega t}\}
</math></center>
<center><math>
V(\mathbf{x},t)=Re \{ v(\mathbf{x},\omega) e^{-i\omega t}\}
</math></center>
and this results in a considerable simplification of the problem. The complex amplitudes of the dynamic quantities vary with frequency and by removing the time-dependence the governing equations and boundary conditions will feature these terms only.
However, \citeasnoun{mciver2003} have shown that the frequency-domain potential can also be defined using the following operations on the time-domain solution:

*take a Laplace transform of the time-domain potential
<center><math>
\hat{\phi}(\mathbf{x},s)=\int^{\infty}_{0}\Phi(\mathbf{x},t)e^{-s t}\, dt ,\quad \textrm{Re } s >0;
</math></center>

*apply the change of variables <math>s=-i\omega</math> so that
<center><math>
\phi(\mathbf{x},\omega)=\hat{\phi}(\mathbf{x},-i\omega).
</math></center>

*noting that <math>\phi(\mathbf{x},-\omega)=\bar{\phi}(\mathbf{x},\omega)</math>, the inverse Fourier transform is given by
<center><math>(3)
\Phi(\mathbf{x},t)=\frac{1}{2\pi}\int^{\infty}_{-\infty}\phi(\mathbf{x},\omega)e^{-i\omega t} d\omega =\frac{1}{\pi}Re\int^{\infty}_{0}\phi(\mathbf{x},\omega)e^{-i\omega t}\,d\omega
</math></center>
where the path of integration must pass over any singularities of <math>\phi</math> that lie on the real axis and it has been assumed that there is no motion prior to <math>t=0</math>, i.e. <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math>.

If <math>\omega</math> is real then <math>\phi(\mathbf{x},\omega)</math> is just the frequency domain potential. By the principle of causality, which precludes the existence of the response before the cause so that <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math> , no singularities can exist in <math>\textrm{Im } \omega>0</math>. Therefore, if it is assumed that no singularities exist on the real axis the inverse transform will integrate the frequency-domain potential over the full range of <math>\omega</math>. With this approach, the frequency-domain equations are obtained by applying the Fourier transform to the time-dependent equations. However, unlike the equations resulting from the assumptions~(2) of time-harmonic motion from <math>t= -\infty</math>, the initial conditions will be present in the frequency-domain equations arising from the Fourier transformation. Although, the latter equations are more mathematically consistent it is usual to set all initial conditions to zero before attempting to solve the equations even if this is not the case. Nevertheless, in the frequency-domain analysis of resonances in coupled motion problems \citeasnoun{pmciver2005} retains the initial condition terms in the equation of motion for the structure and obtains some important results regarding the resonant behaviour of the motion. Therefore, the retention of the initial condition terms depends on the application of the subsequent equations.

Many analytical and approximate solutions exist for frequency-domain problems involving structures with simple geometries. Thus, in the simplest frequency-domain radiation and diffraction problems it is possible to completely determine the velocity potential analytically. The hydrodynamic forces, which are crucial in the context of marine engineering, can then be calculated directly from the potential. For more general geometries this is not possible and the hydrodynamic forces must be computed using numerical methods. Nevertheless, it is useful to consider the analytical expressions for the various potentials and forces in the frequency-domain problem.

In a linearised coupled motion problem, it is customary to decompose the total velocity potential into a scattering potential <math>\phi^{S}</math> and a radiation potential <math>\phi^{R}</math>. Thus, the coupled motion problem will require the solution of the radiation problem and the scattering problem. The radiation potential can be further decomposed into a sum over the six modes of motion so that the total frequency domain potential is
<center><math>
\phi=\phi^{S}+\sum_{\mu}v_{\mu}\phi_{\mu}
</math></center>

where <math>u_{\mu}</math> is the complex amplitude of the generalised velocity in the <math>\mu</math> direction and <math>\phi_{\mu}</math> describes the fluid response due to the forced oscillations in mode <math>\mu</math> with unit velocity amplitude. The velocities in each mode will be determined from the frequency domain equation of motion for the body. However, this requires the computation of the radiation and exciting forces on the structure. Therefore, it is necessary to consider the radiation problem prior to the coupled motion equation. It is first important to note that, for a given mode <math>\mu</math> the boundary condition on the structure will be

<center><math>

\frac{\partial\phi_{\mu}}{\partial n}=n_{\mu},

</math></center>

because the total velocity is

<center><math>

v(\omega)=\sum_{\mu}v_{\mu}n_{\mu}

</math></center>

where <math>n_{\mu}</math> is the <math>\mu</math> component of the generalised normal and <math>v_{\mu}</math> is the component of the generalised complex velocity amplitude in this direction. The hydrodynamic force on the structure in the <math>\mu</math> direction due to the fluid response to the forced oscillations is

<center><math>(5)

F_{\mu}^{R}=i\omega\rho\iint_{S_{B}}\phi^{R}n_{\mu} \,dS

</math></center>

and the force due to the diffraction of an incident wave by the fixed structure is

<center><math>(6)

F_{\mu}^{S}=i\omega\rho\iint_{S_{B}}\phi^{S}n_{\mu} \,dS

</math></center>

where the time-dependence has been removed in both cases. By expressing the radiation potential as a sum over the individual modes, the radiation force can written as <math>\sum_{\nu} v_{\nu}f_{\nu\mu}</math> where

<center><math>

f_{\nu\mu}=i\omega\rho\iint_{\Gamma} \phi_{\nu}n_{\mu} \,dS.

</math></center>

It is conventional to decompose the radiation force into a term featuring the added mass matrix <math>a_{\alpha\beta}</math> and a term featuring the damping matrix <math>b_{\alpha\beta}</math> as follows

<center><math>(7)

f_{\beta\alpha}=i\omega(a_{\beta\alpha}+\frac{ib_{\beta\alpha}}{\omega})

</math></center>

where the added mass and damping coefficients are in phase with the acceleration and velocity respectively. These real quantities depend only on the geometry of the structure and the oscillation frequency <math>\omega</math> and describe many of the properties of the structure. In particular, (as will be shown see later) if the variation of these coefficients with frequency is known then we can predict the resonant behaviour of the structure in the time-domain as well as the frequency domain. The computation of these quantities was considered to be very important in marine engineering and naval architecture and so a variety of different computer codes were developed to compute the coefficients efficiently. The boundary element method forms the basis of most of the computer code algorithms, the most widely-used example being WAMIT (see http:<math>\backslash\backslash</math>www.wamit.com) which determines frequency domain solutions for prescribed structures and structure motions.

==Frequency-domain equation of motion==

The radiation force coefficients also play a significant role in determining the motion of a floating structure possibly subject to mooring and applied forces. To obtain the correct equation of motion in the frequency domain it is important to adopt the approach similar to that taken by [[McIver 2005]], i.e. keep the initial condition terms in the Fourier transform operations. Therefore, if the Fourier transform of the velocity is <math>v_{\mu}(\omega)</math> then the Fourier transform of the acceleration is given by

<center><math>

\int_{0}^{\infty}\ddot{X}_{\mu}(t)e^{i\omega t}dt=-i\omega v_{\mu}(\omega) - V_{\mu}(0)

</math></center>

the time-derivative of the potential obeys a similar relation

<center><math>

\int_{0}^{\infty}\frac{\partial\Phi}{\partial t}e^{i\omega t}dt=-i\omega \phi_{\mu}(\omega) - \Phi(\mathbf{x},0).

</math></center>

Therefore, the ini\partial conditions of the structure <math>(X_{\mu}(0),V_{\mu}(0))</math> and of the potential <math>\Phi(\mathbf{x},0)</math> will be present in the Fourier transform of the equation motion which will be referred to as the frequency-domain equation of motion. If the frequency domain potential is decomposed in the same manner as before then the Fourier transform of the equation of motion for the structure is
<center><math>
M\left[-V_{\mu}(0)-i\omega v_{\mu}(\omega)\right]
=-\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS+F^{S}_{\mu}(\omega)+
i\omega \sum_{\nu} (f_{\mu\nu}(\omega)+i\gamma_{\mu\nu}/\omega)v_{\nu}(\omega)-
\sum_{\nu} c_{\mu\nu}\frac{v(\omega)+X(0)}{-i\omega}+f^{A}_{\mu}(\omega)
</math></center>
for <math>\mu=1,\ldots,6</math>, where <math>f^{A}_{\mu}(\omega)</math> is the Fourier transform of the applied force <math>F_{\mu}(t)</math> in equation~(\ref{linearisedmotion}). Although it is assumed that <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math>, for a non-zero initial velocity <math>\lim_{t^{+}\rightarrow 0}\Phi(\mathbf{x},t)\neq 0</math> because of the impulsive pressure on the free-surface resulting from the initial motion of the pressure. Therefore, the first term on the right-hand side of the above equation is not identically zero, rather is given by
<center><math>
\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}dS=a(\infty)V(0)
</math></center>
[[Mei 1983]]
where <math>a(\infty)=\lim_{\omega\rightarrow\infty}a(\omega)</math> is the infinite frequency added mass.

The frequency-domain equation is usually re-expressed in the following form
<center><math>
\sum_{\nu} \{c_{\mu\nu}-\omega^{2}\left[M_{\mu\mu}\delta_{\mu\nu}+a_{\mu\nu}(\omega)+i(b_{\mu\nu}+\gamma_{\mu\nu})/\omega\right]\}v_{\nu}(\omega)
= -i\omega\left[ F_{\mu}^{S}(\omega)+f_{\mu}(\omega)+(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)\right]-\sum_{\nu}c_{\mu\nu}X_{\nu}(0)
</math></center>
for <math>\mu=1,\ldots,6</math>. If it is assumed that the structure can only move in one mode of motion and that there is no incident wave or applied force then the complex velocity will be given by
<center><math>
v_{\mu}(\omega)=\frac{-i\omega(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)-c_{\mu\mu}X_{\mu}(0)}{c_{\mu\mu}-\omega^{2}\left[M_{\mu\mu}+a_{\mu\mu}(\omega)+i(b_{\mu\mu}+\gamma_{\mu\mu})/\omega\right]}.
</math></center>

As described by [[McIver 2006]], the zero of the denominator gives the location of the pole referred to as the motion resonance. The complex force coefficient <math>f_{\mu\nu}(\omega)</math> will also possess a pole, however this resonance is annulled by the frequency-dependence of the complex velocity (see [[McIver 2006]]) and so the motion, in the absence of incident waves, is dominated by the motion resonance term. Solving for <math>v(\omega)</math> then the time-domain velocity can be recovered using the inverse transform
<center><math>
V(t)=\frac{1}{\pi} Re \int^{\infty}_{0}v(\omega)e^{-i\omega t} dt.
</math></center>

==Integro-differential equation method==

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by <math>\delta(t-\tau)V_{\alpha}(\tau)</math> where <math>\dot{X_{\alpha}}=V_{\alpha}</math>. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function <math>V_{\alpha}(t)</math> an integral of the impulse response over the range <math>-\infty<\tau<\infty</math> must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

<center><math>(11)

\mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau

</math></center>

where <math>m_{\beta\alpha}(\infty)</math> is an infinite frequency added mass coefficient and <math>L_{\beta\alpha}(t-\tau)</math> is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. <math>V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \}</math>, the impulse response function <math>L_{\beta\alpha}</math> can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

<center><math>

\begin{split}

a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, d\tau \

b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, d\tau

\end{split}

</math></center>

and so if the added mass or damping coefficients are known for all frequencies <math>0\leq\omega\leq\infty</math>, the impulse response or memory function <math>L_{\alpha\beta}(t)</math> can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

<center><math>

\left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t),

</math></center>
for <math>\alpha=1,\ldots,6</math>, where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency <math>\omega</math> on the interval <math>[0,\infty)</math>. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

==Generalised eigenfunction expansion method==

The singularity expansion method (SEM) is based on a general scattering theory for general wave problems and has been adapted by \citeasnoun{hazardSEM} and \citeasnoun{meylaniwwwfb2008}

to problems in hydrodynamics and hydro-elasticity. The central concept in the theory is to express the governing equations for the scattering problem in terms of an evolution equation involving a unitary operator and to use the generalised eigenfunctions and eigenfrequencies of this operator to give an approximation to the time-domain solution of the scattering problem. To give a brief overview of the method the scattering equations must be re-expressed using a harmonic lifting operator <math>\mathbf{B}</math> which maps a function defined on the free-surface onto the total fluid domain. This harmonic lifting operator is denoted by <math>\mathbf{B}</math> so that <math>\mathbf{B}\Psi=\Phi</math> where <math>\Psi=\Phi</math> on the free-surface <math>F</math>. If <math>D</math> denotes the total fluid domain, then potential <math>\Phi</math> satisfies

<center><math>\begin{matrix}

\nabla^{2}\Phi&=0 \quad\textrm{in } D, \

\frac{\partial\Phi}{\partial n} &= 0 \quad\textrm{on } \partial D,\

\Phi&=\Psi \quad\textrm{on } F

\end{matrix}</math></center>

where <math>\partial D</math> includes the bed and the structure surface. The Dirichlet to Neumann map <math>\partial_{\mathbf{n}}\mathbf{B}</math> defined by

<center><math>(12)

\partial_{\mathbf{n}}\mathbf{B}\Psi=\frac{\partial\Phi}{\partial z}, \quad x\in F,

</math></center>

then recovers the corresponding normal derivative to <math>\Psi</math> on the free surface.

The evolution equation

<center><math>

i\frac{\partial \mathbf{U}}{\partial t} = \mathcal{A}\mathbf{U}

</math></center>

accounts for the linear time-dependent free-surface equations using the vector

<center><math>

\mathbf{U}= \begin{pmatrix}

\Psi \

-i\eta

\end{pmatrix},

</math></center>

with <math>\eta</math> denoting the time-dependent free-surface elevation, and the operator <math>\mathcal{A}</math> which is both unitary and self-adjoint given by

<center><math>

\mathcal{A}= \begin{pmatrix}

0 & g \

\partial_{\mathbf{n}}\mathbf{B} & 0 \

\end{pmatrix}

</math></center>

A self-adjoint operator will possess an entirely real spectrum and the generalised eigenfunctions <math>\mathbf{u}=(\psi,-i\zeta)^{T}</math>, which are non-trivial solutions of

<center><math>(13)

\mathcal{A}\mathbf{u}=\omega\mathbf{u},

</math></center>

are just frequency-domain scattering solutions. (On a technical note, the word `generalised' precedes eigenfunction because the energy is unbounded.) This can be shown easily by combining the scalar equations resulting from (13) so as to give the frequency-domain free-surface condition. For a given frequency <math>\omega</math>, the free-surface condition in three-dimensions is satisfied by waves from an infinite number of directions and so for each <math>\omega</math> there is an infinite set of eigenfunctions <math>\mathbf{u}_{n}=(1,\omega/g)^{T}\psi_{n}(\mathbf{x},\omega)</math>, with the <math>n^{th}</math> eigenfunction corresponding to incident waves of the form <math>J_{n}(k r) e^{in\theta}</math>. The general solution of the time-evolution equation is, from spectral theory,

<center><math>(14)

\mathbf{U}(\mathbf{x},t)=\int^{\infty}_{-\infty}\left[ \sum_{n}f_{n}(\omega)\mathbf{u}_{n}(\mathbf{x},\omega) \right]e^{-i\omega t}\, d\omega

</math></center>

where <math>f_{n}(\omega)</math> is determined by the initial conditions now expressed as <math>\mathbf{U}(\mathbf{x},0)</math>. To obtain this expression for <math>f_{n}(\omega)</math>, apply the energy inner product to (14) evaluated at <math>t=0</math>. The eigenfunctions satisfy the orthogonality condition

<center><math>(15)

<\mathbf{u}_{m}(\mathbf{x},\omega),\mathbf{u}_{n}(\mathbf{x},\omega')>_{E}=\Lambda_{m}(\omega)\delta_{mn}\delta(\omega-\omega')

</math></center>

where <math><>_{E}</math> denotes a special energy inner product (defined with a <math>\mathcal{H}</math> subscript by \citeasnoun{meylaniwwwfb2008}) and it can be shown that

<center><math>

\Lambda_{m}(\omega)=\frac{4\pi\omega^{2}}{g k}\frac{d\omega}{dk}.

</math></center>

Therefore, it is straightforward to show that

<center><math>

f_{n}(\omega)=\frac{1}{\Lambda_{n}(\omega)}<\mathbf{U}(\mathbf{x},0),\mathbf{u}_{n}(\mathbf{x},\omega)>_{E}.

</math></center>

by evaluating the energy inner product <math><\mathbf{U}(\mathbf{x},0),\mathbf{u}_{n}(\mathbf{x},\omega)>_{E}</math> using the orthogonality relation~(15) and the definition~(14). This expression can be further simplified using the definition of the energy product but details will not be provided here. Instead, it should be noted that an analytic expression for the general solution has been obtained. Thus, for a given scattering problem it is in theory possible to obtain the motion of the free-surface and the potential on the free-surface.

The singularity expansion method involves moving the path of integration in~(14) in the <math>\omega</math>-plane across the singularities in the lower half plane Im<math>\,\omega\leq0</math> using the method of contour integration. Thus, the general solution will consist of contributions from the poles, any branch cuts, the path at infinity and the remainder of the path located below the poles in the <math>\omega</math>-plane. As explained by \citeasnoun{hazardSEM}, the contributions from infinity are assumed to be identically zero and the contribution from the remainder of the path decays faster than the pole contributions. Given that the contribution from the branch cuts are thought to be significant only for very large times an approximation to the general solution can be obtained for medium and large <math>t</math> because the contributions from the poles dominate for this range of times. Therefore, the SEM requires a knowledge of the pole structure of the integrand to approximate the complete integral. Furthermore, the behaviour of the eigenfunctions in the vicinity of the poles must be known in order to compute the residue of each pole. The scattering potential will satisfy

<center><math>

(I+T(\omega))\phi^{S}=g_{I}

</math></center>

where the form of the operator <math>T(\omega)</math> depends on the solution method chosen and <math>g_{I}</math> is determined by the incident wave. Given <math>g_{I}</math>, the scattered field is

<center><math>

\phi^{S}=(I+T(\omega))^{-1}g_{I}

</math></center>

and the pole structure of <math>\phi_{S}</math> will be inherited from the operator <math>(I+T(\omega))^{-1}</math>, referred to as the resolvent.

In the case where the resolvent operator is approximated by a matrix, it can be shown that the potential has the form

<center><math>(16)

\phi_{n}(\mathbf{x},\omega)\sim\frac{\alpha_{nj}v_{j}(\mathbf{x})}{\omega-\omega_{j}} \textrm{ as } \omega\rightarrow\omega_{j}

</math></center>

in the vicinity of the pole. The poles themselves are determined by locating values of <math>\omega</math> in the lower complex plane where the resolvent is not invertible. Thus, <math>v_{j}</math> is a generalised eigenfunction of <math>A_{j0}=I+T(\omega_{j})</math>. The integral in equation~(14) can be approximated by closing the integration path in the lower half plane and then moving the integration path across a finite number of <math>P</math> poles and summing over the contributions of these poles. As described by~\citeasnoun{hazardSEM}, any branch cuts in the complex <math>\omega</math>-plane will only be significant at very large times and the integrals at infinity are expected to be zero. Therefore, the contribution of the <math>P</math> poles closest to the real-<math>\omega</math> axis (and hence with the smallest decay rates <math>e^{-Im(\omega_{j})}</math>) will dominate in the medium term. So, after neglecting other contributions to the integral and using~(16) to determine the residues of the poles, the potential on the free-surface will be given by

<center><math>(17)

\Psi(\mathbf{x},t)\approx -2\pi\sum_{j}\left[ \sum_{n}f_{n}(\omega_{j})\alpha_{nj}\right]v_{j}(\mathbf{x},0)e^{-i\omega_{j}t}.

</math></center>

Although full details of how this expression can be evaluated are not given here, examples of applications of this method are given by~\citeasnoun{meylaniwwwfb2007}, \citeasnoun{meylaniwwwfb2008} and \citeasnoun{meylan2002}. Each of these papers also contain more detailed instructions regarding the computational aspects of the method, such as determining the locations of the poles.

\citeasnoun{hazardSEM} notes that the SEM is a non-rigorous method and requires some heuristic arguments to justify its use. However, in the specific cases considered the results are generally accurate for a large range of times and much frequency-domain information regarding resonances is inherent in the time-domain solution. At present, the method only applies to scattering problems; however, it is hoped that it can be extended to radiation and coupled motion problems. It is not clear how the forces on the structure will be determined.

\f1

}

{\rtf1\ansi\ansicpg1252\deff0\deflang2057{\fonttbl{\f0\fnil\fcharset0 Courier New;}{\f1\fswiss\fcharset0 Arial;}}

{\*\generator Msftedit 5.41.21.2508;}\viewkind4\uc1d\f0\fs20=Review of time-domain models=

==Frequency domain definitions==(1)

Linearised water-wave problems were first investigated in the frequency domain wherein all motions are assumed to be time-harmonic and to have begun from <math>t= -\infty</math> . The time-dependence can be removed in a straightforward manner by substituting for the potential and any structural motions using

<center><math>(2)

\begin{split}

\Phi(\mbf{x},t)=Re \{\phi(\mbf{x},\omega) e^{-i\omega t}\} \

V(\mbf{x},t)=Re \{ v(\mbf{x},\omega) e^{-i\omega t}\}

\end{split}

</math></center>

and this results in a considerable simplification of the problem. The complex amplitudes of the dynamic quantities vary with frequency and by removing the time-dependence the governing equations and boundary conditions will feature these terms only.
However, \citeasnoun{mciver2003} have shown that the frequency-domain potential can also be defined using the following operations on the time-domain solution:

*take a Laplace transform of the time-domain potential
<center><math>
\hat{\phi}(\mbf{x},s)=\int^{\infty}_{0}\Phi(\mbf{x},t)e^{-s t}\, dt ,\quad \textrm{Re } s >0;
</math></center>

*apply the change of variables <math>s=-i\omega</math> so that
<center><math>
\phi(\mbf{x},\omega)=\hat{\phi}(\mbf{x},-i\omega).
</math></center>

*noting that <math>\phi(\mbf{x},-\omega)=\bar{\phi}(\mbf{x},\omega)</math>, the inverse Fourier transform is given by
<center><math>(3)
\Phi(\mbf{x},t)=\frac{1}{2\pi}\int^{\infty}_{-\infty}\phi(\mbf{x},\omega)e^{-i\omega t} d\omega =\frac{1}{\pi}Re\int^{\infty}_{0}\phi(\mbf{x},\omega)e^{-i\omega t}\,d\omega
</math></center>
where the path of integration must pass over any singularities of <math>\phi</math> that lie on the real axis and it has been assumed that there is no motion prior to <math>t=0</math>, i.e. <math>\Phi(\mbf{x},t)=0</math> for <math>t<0</math>.

If <math>\omega</math> is real then <math>\phi(\mbf{x},\omega)</math> is just the frequency domain potential. By the principle of causality, which precludes the existence of the response before the cause so that <math>\Phi(\mbf{x},t)=0</math> for <math>t<0</math> , no singularities can exist in <math>\textrm{Im } \omega>0</math>. Therefore, if it is assumed that no singularities exist on the real axis the inverse transform will integrate the frequency-domain potential over the full range of <math>\omega</math>. With this approach, the frequency-domain equations are obtained by applying the Fourier transform to the time-dependent equations. However, unlike the equations resulting from the assumptions~(2) of time-harmonic motion from <math>t= -\infty</math>, the initial conditions will be present in the frequency-domain equations arising from the Fourier transformation. Although, the latter equations are more mathematically consistent it is usual to set all initial conditions to zero before attempting to solve the equations even if this is not the case. Nevertheless, in the frequency-domain analysis of resonances in coupled motion problems \citeasnoun{pmciver2005} retains the initial condition terms in the equation of motion for the structure and obtains some important results regarding the resonant behaviour of the motion. Therefore, the retention of the initial condition terms depends on the application of the subsequent equations.

Many analytical and approximate solutions exist for frequency-domain problems involving structures with simple geometries. Thus, in the simplest frequency-domain radiation and diffraction problems it is possible to completely determine the velocity potential analytically. The hydrodynamic forces, which are crucial in the context of marine engineering, can then be calculated directly from the potential. For more general geometries this is not possible and the hydrodynamic forces must be computed using numerical methods. Nevertheless, it is useful to consider the analytical expressions for the various potentials and forces in the frequency-domain problem.

In a linearised coupled motion problem, it is customary to decompose the total velocity potential into a scattering potential <math>\phi^{S}</math> and a radiation potential <math>\phi^{R}</math>. Thus, the coupled motion problem will require the solution of the radiation problem and the scattering problem. The radiation potential can be further decomposed into a sum over the six modes of motion so that the total frequency domain potential is

<center><math>(4)

\phi=\phi^{S}+\sum_{\mu}v_{\mu}\phi_{\mu}

</math></center>

where <math>u_{\mu}</math> is the complex amplitude of the generalised velocity in the <math>\mu</math> direction and <math>\phi_{\mu}</math> describes the fluid response due to the forced oscillations in mode <math>\mu</math> with unit velocity amplitude. The velocities in each mode will be determined from the frequency domain equation of motion for the body. However, this requires the computation of the radiation and exciting forces on the structure. Therefore, it is necessary to consider the radiation problem prior to the coupled motion equation. It is first important to note that, for a given mode <math>\mu</math> the boundary condition on the structure will be

<center><math>

\frac{tial\phi_{\mu}}{tial n}=n_{\mu},

</math></center>

because the total velocity is

<center><math>

v(\omega)=\sum_{\mu}v_{\mu}n_{\mu}

</math></center>

where <math>n_{\mu}</math> is the <math>\mu</math> component of the generalised normal and <math>v_{\mu}</math> is the component of the generalised complex velocity amplitude in this direction. The hydrodynamic force on the structure in the <math>\mu</math> direction due to the fluid response to the forced oscillations is

<center><math>(5)

F_{\mu}^{R}=i\omega\rho\iint_{S_{B}}\phi^{R}n_{\mu} \,dS

</math></center>

and the force due to the diffraction of an incident wave by the fixed structure is

<center><math>(6)

F_{\mu}^{S}=i\omega\rho\iint_{S_{B}}\phi^{S}n_{\mu} \,dS

</math></center>

where the time-dependence has been removed in both cases. By expressing the radiation potential as a sum over the individual modes, the radiation force can written as <math>\sum_{\nu} v_{\nu}f_{\nu\mu}</math> where

<center><math>

f_{\nu\mu}=i\omega\rho\iint_{\Gamma} \phi_{\nu}n_{\mu} \,dS.

</math></center>

It is conventional to decompose the radiation force into a term featuring the added mass matrix <math>a_{\alpha\beta}</math> and a term featuring the damping matrix <math>b_{\alpha\beta}</math> as follows

<center><math>(7)

f_{\beta\alpha}=i\omega(a_{\beta\alpha}+\frac{ib_{\beta\alpha}}{\omega})

</math></center>

where the added mass and damping coefficients are in phase with the acceleration and velocity respectively. These real quantities depend only on the geometry of the structure and the oscillation frequency <math>\omega</math> and describe many of the properties of the structure. In particular, (as will be shown see later) if the variation of these coefficients with frequency is known then we can predict the resonant behaviour of the structure in the time-domain as well as the frequency domain. The computation of these quantities was considered to be very important in marine engineering and naval architecture and so a variety of different computer codes were developed to compute the coefficients efficiently. The boundary element method forms the basis of most of the computer code algorithms, the most widely-used example being WAMIT (see http:<math>\backslash\backslash</math>www.wamit.com) which determines frequency domain solutions for prescribed structures and structure motions.

==Frequency-domain equation of motion==(8)

The radiation force coefficients also play a significant role in determining the motion of a floating structure possibly subject to mooring and applied forces. To obtain the correct equation of motion in the frequency domain it is important to adopt the approach similar to that taken by~\citeasnoun{pmciver2005}, i.e. keep the initial condition terms in the Fourier transform operations. Therefore, if the Fourier transform of the velocity is <math>v_{\mu}(\omega)</math> then the Fourier transform of the acceleration is given by

<center><math>

\int_{0}^{\infty}\ddot{X}_{\mu}(t)e^{i\omega t}dt=-i\omega v_{\mu}(\omega) - V_{\mu}(0)

</math></center>

the time-derivative of the potential obeys a similar relation

<center><math>

\int_{0}^{\infty}\frac{tial\Phi}{tial t}e^{i\omega t}dt=-i\omega \phi_{\mu}(\omega) - \Phi(\mbf{x},0).

</math></center>

Therefore, the initial conditions of the structure <math>(X_{\mu}(0),V_{\mu}(0))</math> and of the potential <math>\Phi(\mbf{x},0)</math> will be present in the Fourier transform of the equation motion which will be referred to as the frequency-domain equation of motion. If the frequency domain potential is decomposed in the same manner as before~(4) then the Fourier transform of the equation of motion~(\ref{linearisedmotion}) for the structure is

<center><math>(9)

\begin{split}

M\left[-V_{\mu}(0)-i\omega v_{\mu}(\omega)\right]&

=-\rho\iint_{\Gamma}(-\Phi(\mbf{x},0))n_{\mu}dS+F^{S}_{\mu}(\omega)+\

i\omega \sum_{\nu}&(f_{\mu\nu}(\omega)+i\gamma_{\mu\nu}/\omega)v_{\nu}(\omega)-

\sum_{\nu} c_{\mu\nu}\frac{v(\omega)+X(0)}{-i\omega}+f^{A}_{\mu}(\omega)

\end{split}

</math></center>

for <math>\mu=1,\ldots,6</math>, where <math>f^{A}_{\mu}(\omega)</math> is the Fourier transform of the applied force <math>F_{\mu}(t)</math> in equation~(\ref{linearisedmotion}). Although it is assumed that <math>\Phi(\mbf{x},t)=0</math> for <math>t<0</math>, for a non-zero initial velocity <math>\lim_{t^{+}\rightarrow 0}\Phi(\mbf{x},t)\neq 0</math> because of the impulsive pressure on the free-surface resulting from the initial motion of the pressure. Therefore, the first term on the right-hand side of the above equation is not identically zero, rather is given by

<center><math>

\rho\iint_{\Gamma}(-\Phi(\mbf{x},0))n_{\mu}dS=a(\infty)V(0)

</math></center>

(see \citeasnoun{ccmei2}, \S~8.12.1)

where <math>a(\infty)=\lim_{\omega\rightarrow\infty}a(\omega)</math> is the infinite frequency added mass.

The frequency-domain equation is usually re-expressed in the following form

<center><math>

\begin{split}

\sum_{\nu}&\{c_{\mu\nu}-\omega^{2}\left[M_{\mu\mu}\delta_{\mu\nu}+a_{\mu\nu}(\omega)+i(b_{\mu\nu}+\gamma_{\mu\nu})/\omega\right]\}v_{\nu}(\omega) \

=& -i\omega\left[ F_{\mu}^{S}(\omega)+f_{\mu}(\omega)+(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)\right]-\sum_{\nu}c_{\mu\nu}X_{\nu}(0)\end{split}

</math></center>

for <math>\mu=1,\ldots,6</math>. If it is assumed that the structure can only move in one mode of motion and that there is no incident wave or applied force then the complex velocity will be given by

<center><math>(10)

v_{\mu}(\omega)=\frac{-i\omega(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)-c_{\mu\mu}X_{\mu}(0)}{c_{\mu\mu}-\omega^{2}\left[M_{\mu\mu}+a_{\mu\mu}(\omega)+i(b_{\mu\mu}+\gamma_{\mu\mu})/\omega\right]}.

</math></center>

As described by \citeasnoun{mciver2006}, the zero of the denominator gives the location of the pole referred to as the motion resonance. The complex force coefficient <math>f_{\mu\nu}(\omega)</math> will also possess a pole, however this resonance is annulled by the frequency-dependence of the complex velocity (see \S~6 of \citeasnoun{mciver2006}) and so the motion, in the absence of incident waves, is dominated by the motion resonance term. Solving equation~(10) for <math>v(\omega)</math> then the time-domain velocity can be recovered using the inverse transform

<center><math>

V(t)=\frac{1}{\pi} Re \int^{\infty}_{0}v(\omega)e^{-i\omega t} dt.

</math></center>

==Integro-differential equation method==

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by <math>\delta(t-\tau)V_{\alpha}(\tau)</math> where <math>\dot{X_{\alpha}}=V_{\alpha}</math>. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function <math>V_{\alpha}(t)</math> an integral of the impulse response over the range <math>-\infty<\tau<\infty</math> must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

<center><math>(11)

\mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau

</math></center>

where <math>m_{\beta\alpha}(\infty)</math> is an infinite frequency added mass coefficient and <math>L_{\beta\alpha}(t-\tau)</math> is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. <math>V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \}</math>, the impulse response function <math>L_{\beta\alpha}</math> can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

<center><math>

\begin{split}

a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, d\tau \

b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, d\tau

\end{split}

</math></center>

and so if the added mass or damping coefficients are known for all frequencies <math>0\leq\omega\leq\infty</math>, the impulse response or memory function <math>L_{\alpha\beta}(t)</math> can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

<center><math>

\left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,d\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t),

</math></center>
for <math>\alpha=1,\ldots,6</math>, where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency <math>\omega</math> on the interval <math>[0,\infty)</math>. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

==Generalised eigenfunction expansion method==

The singularity expansion method (SEM) is based on a general scattering theory for general wave problems and has been adapted by \citeasnoun{hazardSEM} and \citeasnoun{meylaniwwwfb2008}

to problems in hydrodynamics and hydro-elasticity. The central concept in the theory is to express the governing equations for the scattering problem in terms of an evolution equation involving a unitary operator and to use the generalised eigenfunctions and eigenfrequencies of this operator to give an approximation to the time-domain solution of the scattering problem. To give a brief overview of the method the scattering equations must be re-expressed using a harmonic lifting operator <math>\mbf{B}</math> which maps a function defined on the free-surface onto the total fluid domain. This harmonic lifting operator is denoted by <math>\mbf{B}</math> so that <math>\mbf{B}\Psi=\Phi</math> where <math>\Psi=\Phi</math> on the free-surface <math>F</math>. If <math>D</math> denotes the total fluid domain, then potential <math>\Phi</math> satisfies

<center><math>\begin{matrix}

\nabla^{2}\Phi&=0 \quad\textrm{in } D, \

\frac{tial\Phi}{tial n} &= 0 \quad\textrm{on } tial D,\

\Phi&=\Psi \quad\textrm{on } F

\end{matrix}</math></center>

where <math>tial D</math> includes the bed and the structure surface. The Dirichlet to Neumann map <math>tial_{\mbf{n}}\mbf{B}</math> defined by

<center><math>(12)

tial_{\mbf{n}}\mbf{B}\Psi=\frac{tial\Phi}{tial z}, \quad x\in F,

</math></center>

then recovers the corresponding normal derivative to <math>\Psi</math> on the free surface.

The evolution equation

<center><math>

i\frac{tial \mbf{U}}{tial t} = \mathcal{A}\mbf{U}

</math></center>

accounts for the linear time-dependent free-surface equations using the vector

<center><math>

\mbf{U}= \begin{pmatrix}

\tab \Psi \

\tab -i\eta

\end{pmatrix},

</math></center>

with <math>\eta</math> denoting the time-dependent free-surface elevation, and the operator <math>\mathcal{A}</math> which is both unitary and self-adjoint given by

<center><math>

[[Category:Linear Water-Wave Theory]]

{{complete page}}

Generalized Eigenfunction Expansion for Water Waves for a Fixed Body

2009-08-17T02:04:27Z

Cfitzgerald: /* Expansion in Eigenfunctions */

== Introduction ==

The generalized eigenfunction method goes back to
the work of [[Povzner 1953]] and [[Ikebe 1960]]. The generalized eigenfunction
method has been applied to water-wave problems by
[[Friedman and Shinbrot 1967]], [[Hazard_Lenoir2002a | Hazard and Lenoir 2002]]
(for the case of a rigid body in water of infinite depth)
and [[Meylan 2002b| Meylan 2002 ]] (for
a thin plate on water of shallow draft). We will present here the theory for a rigid body
in water of finite depth.
rms of this solutions (which we call the generalised
eigenfunctions) because they solve for

The generalized eigenfunction method is based on an inner product in
which the evolution operator is self-adjoint. It follows from the
self-adjointness that we can expand the solution in the
eigenfunctions of the operator. These eigenfunctions
are nothing more than the single-frequency solutions. The main
difficulty is that the eigenfunctions are associated with a continuous
spectrum, and this requires that they be carefully normalized. Once
this is done, we can derive simple expressions which allow the
time-domain problem to be solved in terms of the single-frequency
solutions. The mathematical ideas are discussed in detail in
[[Hazard and Loret 2007]].

{{Equations for fixed bodies in the time domain}}

== Single frequency equations ==

The single frequency solution is based on the assumption that all
time-dependence is given by <math>\mathrm{e}^{\mathrm{i}\omega t}</math> and that the system is
excited by an incident wave. We can then write
<center><math>
\Phi\left( \mathbf{x},t\right) ={\Phi}_\kappa\left( \mathbf{x},\omega\right)
\mathrm{e}^{-\mathrm{i}\omega t},\ \ \ \mathrm{and} \ \ \ \zeta\left(
x,t\right) ={\zeta}_\kappa\left( x,\omega\right) \mathrm{e}^{-\mathrm{i}\omega
t},
</math></center>
where <math>\kappa=1</math> for waves excited by an incident wave from the left
and <math>-1</math> for waves incident from the right.
Under these assumptions, the equations become
<center><math>
\Delta{\Phi}_\kappa\left( \mathbf{x,}\omega\right) =0,\ \ \mathbf{x}\in
\Omega,
</math></center>
<center><math>
\partial_{n}{\Phi}_\kappa=0,\ \ \mathbf{x}\in\partial\Omega,
</math></center>
<center><math>
\partial_{n}{\Phi}_\kappa=0,\ \ z=-h,
</math></center>
<center><math>
-\mathrm{i}\omega{\zeta}_\kappa=\partial_{n}{\Phi}_\kappa,\ \ z=0,\ x\in F,
</math></center>
<center><math>
{\zeta}_\kappa = \mathrm{i}\omega{\Phi}_\kappa,\ \ z=0,\ x\in F.
</math></center>

We must also specify radiations conditions, which are given
by
<center><math>
{\Phi}_1=\frac{1}{\mathrm{i} \omega} \left(
\mathrm{e}^{\mathrm{i} kx} + R_1 e^{-\mathrm{i} k x} \right)\frac{\cosh k\left( z+h\right)
}{\cosh kh},
\,\,\,\mathrm{as}\,\,x\to-\infty,
</math></center>
<center><math>
{\Phi}_1=\frac{1}{\mathrm{i} \omega}
T_1\mathrm{e}^{\mathrm{i} kx}\frac{\cosh k\left( z+h\right) }{\cosh kh},
\,\,\,\mathrm{as}\,\,x\to\infty,
</math></center>
<center><math>
{\Phi}_{-1}=\frac{1}{\mathrm{i} \omega}
T_{-1}\mathrm{e}^{-\mathrm{i} kx}
\frac{\cosh k\left( z+h\right) }{\cosh kh},
\,\,\,\mathrm{as}\,\,x\to\infty,
</math></center>
<center><math>
{\Phi}_{-1}=\frac{1}{\mathrm{i} \omega}
\left(
\mathrm{e}^{-\mathrm{i} kx} + R_{-1} e^{\mathrm{i} k x} \right)\frac{\cosh k\left( z+h\right)
}{\cosh kh},
\,\,\,\mathrm{as}\,\,x\to-\infty.
</math></center>

Note that <math>R_{\kappa}</math> and <math>T_{\kappa}</math> are the reflection and
transmission coefficients respectively and that we have normalized so
that the amplitude (in displacement) is unity. The wavenumber <math>k</math> is
the positive real solution of the [[Dispersion Relation for a Free Surface|dispersion equation]]
<math>k\tanh kh=\omega^{2}</math>,
and we will consider both <math>k(\omega)</math> and <math>\omega(k)</math>
in what follows as required. The solution of the single-frequency
equation may be computationally challenging and for the generalized
eigenfunction expansion the major numerical work is to determine the
single-frequency solutions.

== Time domain calculations ==

The solution in the frequency domain can be used to construct the
solution in the time domain. This is well-known for the case of a
plane incident wave which is initially far from the body, and in this
case the solution can be calculated straightforwardly using the standard Fourier
transform. However, when we consider an initial displacement which is
non-zero around the bodies, we cannot express the time-dependent
solution in terms of the single frequency solutions by a standard Fourier
transform.

We begin with the equations in the time domain subject to the initial conditions given by
Denoting the potential at the surface by
<center><math>
\phi(x,t)=\left.\Phi\left(\mathbf{x},t\right)\right|_{z=0},
</math></center>
we introduce the operator <math>\mathbf{G}</math> which maps the surface
potential to the potential throughout the fluid domain. The operator
<math>\mathbf{G}\psi</math> is found by solving
<center><math>\begin{matrix}
\Delta\Psi\left( \mathbf{x}\right) & = 0,\ \ \mathbf{x}\in\Omega,\\
\partial_{n}\Psi & = 0,\ \ \mathbf{x}\in\partial\Omega,\\
\partial_{n}\Psi & = 0,\ \ z=-h,\\
\Psi & =\psi,\ \ z = 0,\ x\in F,
\end{matrix}</math></center>
and is defined by <math>\mathbf{G}\psi=\Psi.</math> The operator
<math>\partial_{n}\mathbf{G}</math>, which maps the surface potential to the normal
derivative of potential at the surface (called the
Dirichlet-to-Neumann map) is given by
<center><math>
\partial_{n}\mathbf{G}\psi=\left. \partial_{n}\Psi\right\vert _{z=0}.
</math></center>
Therefore the equations in the time domain can be written as
<center><math>
\partial_{t}^2 \zeta + \partial_{n}\mathbf{G} \zeta = 0.
</math></center>
where we can recover the potential using the operator <math>\mathbf{G}</math>.
The evolution operator <math>\partial_{n}\mathbf{G}</math> is
symmetric in the Hilbert space given by the following inner product
<center><math>
\left\langle \zeta,\eta \right\rangle _{\mathcal{H}}=
\int_{F}\zeta \eta ^{*}\,\mathrm{d} x,
</math></center>
where <math> *</math> denotes complex conjugate, and we assume that this
symmetry implies that the operator is self-adjoint. We can prove the
symmetry by using Green's second identity
<center><math>\begin{matrix}
\int_{F}\left( \partial_{n}\mathbf{G}\zeta \right) \left(
\eta\right) ^{*}\,\mathrm{d} x
&=\int_{F}\left( \partial_{n}\mathbf{G}\partial_t\phi \right) \left(
\partial_t\psi\right) ^{*}\,\mathrm{d} x \\
&=\int_{F}\left( \partial_t\phi \right) \left(
\partial_{n}\mathbf{G} \partial_t\psi\right) ^{*}\,\mathrm{d} x
\\
&=\int_{F}\left( \zeta \right) \left(
\partial_{n}\mathbf{G}\eta\right) ^{*}\,\mathrm{d} x,
\end{matrix}</math></center>
where <math>\phi</math> and <math>\psi</math> are the surface potentials associated with
<math>\zeta</math> and <math>\eta</math> respectively.

== Eigenfunctions of <math>\partial_{n}\mathbf{G}</math> ==

The eigenfunctions of <math> \partial_{n}\mathbf{G}</math> satisfy
<center><math>
\partial_{n}\mathbf{G} \zeta = \omega^2 \zeta.
</math></center>

To solve
for the eigenfunctions of <math>\partial_{n}\mathbf{G}</math> we need to solve
the frequency-domain equations, and the radian frequency <math>\omega</math> is exactly
the eigenvalue. To actually calculate the eigenfunctions of
<math>\partial_{n}\mathbf{G}</math> we need to specify the incident wave
potential, and for each frequency we have two eigenfunctions (waves
incident from the left and from the right). It is possible for there
to exist point spectra for this operator which correspond to the
existence of a [[Trapped Modes|trapped mode]] [[McIver 1996]]
and the presence of a trapped mode requires that
the generalized eigenfunction expansion we derive must be modified.

==Normalization of the Eigenfunctions==

The eigenfunctions of <math>\partial_{n}\mathbf{G}</math> (with eigenvalue
<math>\omega</math>) are denoted by <math> \zeta_{\kappa}(x,k\left( \omega\right) ) </math>.
As mentioned previously, determining <math>\zeta_\kappa</math> is the major
computation of the generalized eigenfunction method, but we simply
assume that they are known. We know that the eigenfunctions are
orthogonal for different <math>\omega</math> (from the self-adjointness of
<math>\partial_n\mathbf{G}</math>), and that the waves incident from the left and right
with the same <math>\omega</math> are orthogonal from the identity
<center><math>
R_1 T_{-1}^{*} + R_{-1}^{*} T_{1}=0,
</math></center>
[[Mei 1983]].
It therefore follows that
<center><math>
\left\langle \left( {\zeta}_{\kappa}(x,k\left( \omega_{1}\right) )\right)
,{\zeta}_{\kappa^{\prime}}(x,k\left( \omega_{2}\right) )\right\rangle _{\mathcal{H}
}=\Lambda_{n}\left( \omega_{1}\right) \delta\left( \omega_{1}-\omega
_{2}\right) \delta_{\kappa\kappa^{\prime}},
</math></center>
but we need to determine the normalizing function
<math>\Lambda_{n}\left( \omega_{n}\right)</math>. This is achieved by using the result that the
eigenfunctions satisfy the same normalizing condition with and without
the scatterers present. This result, the proof of which is quite
technical, is well-known and has been shown for many different
situations. The original proof was for Schr\"odinger's equation and was
due to [[Povzner 1953]] and [[Ikebe 1960]]. A proof for the case of Helmholtz
equation was given by [[Wilcox 1975]]. Recently the proof was given
for water waves by [[Hazard and Lenoir 2002]] and [[Hazard and Loret 2007]].

Since the eigenfunctions satisfy the same normalizing condition with
and without the scatterers, we normalize with the scatterers absent.
This means that the eigenfunctions are simply the incident waves, and
the free surface <math>F</math> is the entire axis. This allows us to derive
<center><math>\begin{matrix}
\left\langle \left( {\zeta}_{\kappa}(x,k\left( \omega_{1}\right) )\right)
,{\zeta}_{\kappa^{\prime}}(x,k\left( \omega_{2}\right) )\right\rangle _{\mathcal{H}}
&= \int_{\mathbb{R}}\left( e^{\kappa\mathrm{i} k_{1}
x}\right) \left( e^{\kappa^{\prime}\mathrm{i} k_{2}x}\right)
^{*}\,\mathrm{d} x \\
& =2\pi \delta_{\kappa\kappa^{\prime}}\delta\left( k_{1}-k_{2}\right) \\
& =2\pi\delta_{\kappa\kappa^{\prime}}\delta\left( \omega_{1}-\omega_{2}\right)
\left. \frac{d\omega}{dk}\right\vert _{\omega=\omega_{1}}.
\end{matrix}</math></center>
This result allows us to calculate the time-dependent solution in the
eigenfunctions (or single-frequency solutions).

==Expansion in Eigenfunctions==

We expand the solution for the displacement in the time domain as
<center><math>
\zeta\left( x,t\right)
=\int_{\mathbb{R}^{+}} \sum_{\kappa\in\left\{ -1,1\right\}}
\left\{
f_{\kappa}\left( \omega\right)
\cos(\omega t)+
g_{\kappa}\left( \omega\right)
\frac{\sin(\omega t)}{\omega}\right\}
\zeta_{\kappa}(x,k) d\omega,
</math></center>
where <math>f_\kappa</math> and <math>g_\kappa</math> will be determined from the initial
conditions. Note that here, and in subsequent equations, we are
assuming that <math>k</math> is a function of <math>\omega</math> or that <math>\omega</math> is a
function of <math>k</math> as required. If we take the inner product with
respect to the eigenfunctions <math>\zeta_\kappa</math> we obtain
<center><math>
\left\langle
\zeta_0\left( x\right)
,\zeta_{\kappa}(x,k)\right\rangle _{\mathcal{H}}=2\pi f_{\kappa}\left(
\omega\right) \frac{d\omega}{dk},
</math></center>
and
<center><math>
\left\langle
\partial_t\zeta_0\left( x\right)
,\zeta_{\kappa}(x,k)\right\rangle _{\mathcal{H}}=2\pi g_{\kappa}\left(
\omega\right) \frac{d\omega}{dk}.
</math></center>
We can therefore write, changing the
variable of integration to <math>k</math>, as
<center><math>
\zeta\left( x,t\right)
=\frac{1}{2\pi}\int_{\mathbb{R}^{+}} \sum_{\kappa\in\left\{ -1,1\right\}}
\Big\{
\left\langle \zeta_0\left( x\right)
,\zeta_{\kappa}(x,k)\right\rangle _{\mathcal{H}}
\cos(\omega t)
+ \left\langle
\partial_t \zeta_0\left( x\right)
,\zeta_{\kappa}(x,k)\right\rangle _{\mathcal{H}}
\frac{\sin(\omega t)}{\omega}\Big\}
\zeta_{\kappa}(x,k) dk,
</math></center>

If we take the case when <math>\partial_t\zeta_0( x) =0</math> and
write the integral given by the inner product explicitly, we obtain
<center><math>
\zeta\left( x,t\right) =\int_{\mathbb{R}^{+}}\Big\{ \sum_{\kappa\in\left\{
-1,1\right\} }\left( \frac{1}{2\pi}\int_{F}\zeta_0\left( x^{\prime}\right)
\zeta_{\kappa}(x^{\prime},k) ^{*}\,\mathrm{d} x^{\prime}\right)
\zeta_{\kappa}(x,k)\Big\}\cos(\omega t)dk.
</math></center>

If the scattering structure supports a trapped mode at a particular frequency <math>\omega_{0}</math> then the expression for the free-surface elevation becomes
<center><math>
\zeta\left( x,t\right) =\int_{\mathbb{R}^{+}}\Big\{ \sum_{\kappa\in\left\{
-1,1\right\} }\left( \frac{1}{2\pi}\int_{F}\zeta_0\left( x^{\prime}\right)
\zeta_{\kappa}(x^{\prime},k) ^{*}\,\mathrm{d} x^{\prime}\right)
\zeta_{\kappa}(x,k)\Big\}\cos(\omega t)dk + \left(\frac{\int_{F}\zeta_{0}(x^{\prime})\tilde\zeta(x^{\prime})^{*}}{\int_{F}\tilde\zeta(x^{\prime})\tilde\zeta(x^{\prime})}dx^{\prime} \right)\cos(\omega_{0} t)
</math></center>
where <math>\tilde\zeta(x^{\prime})</math> is the trapped mode free-surface elevation.

==An identity linking waves from the left and right==

A consequence of the requirement that the displacement be real, if the
initial displacement and initial derivative of displacement is real, is
that
<center><math>
\sum_{\kappa\in\left\{
-1,1\right\}} \left\langle \zeta_0\left( x\right)
,\zeta_{\kappa}(x,k)\right\rangle _{\mathcal{H}} \zeta_{\kappa}(x,k),
</math></center>
must be purely real. This can only be true if
<center><math>
\Im \left\{ \zeta_{1}(x^\prime,k)^{*} \zeta_{1}(x,k) \right\}
=
- \Im \left\{ \zeta_{-1}(x^\prime,k)^{*} \zeta_{-1}(x,k) \right\},
\,\,\,x,x^{\prime} \in F.
</math></center>

[[Category:Time-Dependent Linear Water Waves]]

Generalized Eigenfunction Expansion for Water Waves for a Fixed Body

2009-08-17T01:44:33Z

Cfitzgerald: /* Expansion in Eigenfunctions */

Generalized Eigenfunction Expansion for Water Waves for a Fixed Body

2009-08-17T01:37:16Z

Cfitzgerald: /* Expansion in Eigenfunctions */

== Introduction ==

The generalized eigenfunction method goes back to
the work of [[Povzner 1953]] and [[Ikebe 1960]]. The generalized eigenfunction
method has been applied to water-wave problems by
[[Friedman and Shinbrot 1967]], [[Hazard_Lenoir2002a | Hazard and Lenoir 2002]]
(for the case of a rigid body in water of infinite depth)
and [[Meylan 2002b| Meylan 2002 ]] (for
a thin plate on water of shallow draft). We will present here the theory for a rigid body
in water of finite depth.
rms of this solutions (which we call the generalised
eigenfunctions) because they solve for

The generalized eigenfunction method is based on an inner product in
which the evolution operator is self-adjoint. It follows from the
self-adjointness that we can expand the solution in the
eigenfunctions of the operator. These eigenfunctions
are nothing more than the single-frequency solutions. The main
difficulty is that the eigenfunctions are associated with a continuous
spectrum, and this requires that they be carefully normalized. Once
this is done, we can derive simple expressions which allow the
time-domain problem to be solved in terms of the single-frequency
solutions. The mathematical ideas are discussed in detail in
[[Hazard and Loret 2007]].

{{Equations for fixed bodies in the time domain}}

== Single frequency equations ==

The single frequency solution is based on the assumption that all
time-dependence is given by <math>\mathrm{e}^{\mathrm{i}\omega t}</math> and that the system is
excited by an incident wave. We can then write
<center><math>
\Phi\left( \mathbf{x},t\right) ={\Phi}_\kappa\left( \mathbf{x},\omega\right)
\mathrm{e}^{-\mathrm{i}\omega t},\ \ \ \mathrm{and} \ \ \ \zeta\left(
x,t\right) ={\zeta}_\kappa\left( x,\omega\right) \mathrm{e}^{-\mathrm{i}\omega
t},
</math></center>
where <math>\kappa=1</math> for waves excited by an incident wave from the left
and <math>-1</math> for waves incident from the right.
Under these assumptions, the equations become
<center><math>
\Delta{\Phi}_\kappa\left( \mathbf{x,}\omega\right) =0,\ \ \mathbf{x}\in
\Omega,
</math></center>
<center><math>
\partial_{n}{\Phi}_\kappa=0,\ \ \mathbf{x}\in\partial\Omega,
</math></center>
<center><math>
\partial_{n}{\Phi}_\kappa=0,\ \ z=-h,
</math></center>
<center><math>
-\mathrm{i}\omega{\zeta}_\kappa=\partial_{n}{\Phi}_\kappa,\ \ z=0,\ x\in F,
</math></center>
<center><math>
{\zeta}_\kappa = \mathrm{i}\omega{\Phi}_\kappa,\ \ z=0,\ x\in F.
</math></center>

We must also specify radiations conditions, which are given
by
<center><math>
{\Phi}_1=\frac{1}{\mathrm{i} \omega} \left(
\mathrm{e}^{\mathrm{i} kx} + R_1 e^{-\mathrm{i} k x} \right)\frac{\cosh k\left( z+h\right)
}{\cosh kh},
\,\,\,\mathrm{as}\,\,x\to-\infty,
</math></center>
<center><math>
{\Phi}_1=\frac{1}{\mathrm{i} \omega}
T_1\mathrm{e}^{\mathrm{i} kx}\frac{\cosh k\left( z+h\right) }{\cosh kh},
\,\,\,\mathrm{as}\,\,x\to\infty,
</math></center>
<center><math>
{\Phi}_{-1}=\frac{1}{\mathrm{i} \omega}
T_{-1}\mathrm{e}^{-\mathrm{i} kx}
\frac{\cosh k\left( z+h\right) }{\cosh kh},
\,\,\,\mathrm{as}\,\,x\to\infty,
</math></center>
<center><math>
{\Phi}_{-1}=\frac{1}{\mathrm{i} \omega}
\left(
\mathrm{e}^{-\mathrm{i} kx} + R_{-1} e^{\mathrm{i} k x} \right)\frac{\cosh k\left( z+h\right)
}{\cosh kh},
\,\,\,\mathrm{as}\,\,x\to-\infty.
</math></center>

Note that <math>R_{\kappa}</math> and <math>T_{\kappa}</math> are the reflection and
transmission coefficients respectively and that we have normalized so
that the amplitude (in displacement) is unity. The wavenumber <math>k</math> is
the positive real solution of the [[Dispersion Relation for a Free Surface|dispersion equation]]
<math>k\tanh kh=\omega^{2}</math>,
and we will consider both <math>k(\omega)</math> and <math>\omega(k)</math>
in what follows as required. The solution of the single-frequency
equation may be computationally challenging and for the generalized
eigenfunction expansion the major numerical work is to determine the
single-frequency solutions.

== Time domain calculations ==

The solution in the frequency domain can be used to construct the
solution in the time domain. This is well-known for the case of a
plane incident wave which is initially far from the body, and in this
case the solution can be calculated straightforwardly using the standard Fourier
transform. However, when we consider an initial displacement which is
non-zero around the bodies, we cannot express the time-dependent
solution in terms of the single frequency solutions by a standard Fourier
transform.

We begin with the equations in the time domain subject to the initial conditions given by
Denoting the potential at the surface by
<center><math>
\phi(x,t)=\left.\Phi\left(\mathbf{x},t\right)\right|_{z=0},
</math></center>
we introduce the operator <math>\mathbf{G}</math> which maps the surface
potential to the potential throughout the fluid domain. The operator
<math>\mathbf{G}\psi</math> is found by solving
<center><math>\begin{matrix}
\Delta\Psi\left( \mathbf{x}\right) & = 0,\ \ \mathbf{x}\in\Omega,\\
\partial_{n}\Psi & = 0,\ \ \mathbf{x}\in\partial\Omega,\\
\partial_{n}\Psi & = 0,\ \ z=-h,\\
\Psi & =\psi,\ \ z = 0,\ x\in F,
\end{matrix}</math></center>
and is defined by <math>\mathbf{G}\psi=\Psi.</math> The operator
<math>\partial_{n}\mathbf{G}</math>, which maps the surface potential to the normal
derivative of potential at the surface (called the
Dirichlet-to-Neumann map) is given by
<center><math>
\partial_{n}\mathbf{G}\psi=\left. \partial_{n}\Psi\right\vert _{z=0}.
</math></center>
Therefore the equations in the time domain can be written as
<center><math>
\partial_{t}^2 \zeta + \partial_{n}\mathbf{G} \zeta = 0.
</math></center>
where we can recover the potential using the operator <math>\mathbf{G}</math>.
The evolution operator <math>\partial_{n}\mathbf{G}</math> is
symmetric in the Hilbert space given by the following inner product
<center><math>
\left\langle \zeta,\eta \right\rangle _{\mathcal{H}}=
\int_{F}\zeta \eta ^{*}\,\mathrm{d} x,
</math></center>
where <math> *</math> denotes complex conjugate, and we assume that this
symmetry implies that the operator is self-adjoint. We can prove the
symmetry by using Green's second identity
<center><math>\begin{matrix}
\int_{F}\left( \partial_{n}\mathbf{G}\zeta \right) \left(
\eta\right) ^{*}\,\mathrm{d} x
&=\int_{F}\left( \partial_{n}\mathbf{G}\partial_t\phi \right) \left(
\partial_t\psi\right) ^{*}\,\mathrm{d} x \\
&=\int_{F}\left( \partial_t\phi \right) \left(
\partial_{n}\mathbf{G} \partial_t\psi\right) ^{*}\,\mathrm{d} x
\\
&=\int_{F}\left( \zeta \right) \left(
\partial_{n}\mathbf{G}\eta\right) ^{*}\,\mathrm{d} x,
\end{matrix}</math></center>
where <math>\phi</math> and <math>\psi</math> are the surface potentials associated with
<math>\zeta</math> and <math>\eta</math> respectively.

== Eigenfunctions of <math>\partial_{n}\mathbf{G}</math> ==

The eigenfunctions of <math> \partial_{n}\mathbf{G}</math> satisfy
<center><math>
\partial_{n}\mathbf{G} \zeta = \omega^2 \zeta.
</math></center>

To solve
for the eigenfunctions of <math>\partial_{n}\mathbf{G}</math> we need to solve
the frequency-domain equations, and the radian frequency <math>\omega</math> is exactly
the eigenvalue. To actually calculate the eigenfunctions of
<math>\partial_{n}\mathbf{G}</math> we need to specify the incident wave
potential, and for each frequency we have two eigenfunctions (waves
incident from the left and from the right). It is possible for there
to exist point spectra for this operator which correspond to the
existence of a [[Trapped Modes|trapped mode]] [[McIver 1996]]
and the presence of a trapped mode requires that
the generalized eigenfunction expansion we derive must be modified.

==Normalization of the Eigenfunctions==

The eigenfunctions of <math>\partial_{n}\mathbf{G}</math> (with eigenvalue
<math>\omega</math>) are denoted by <math> \zeta_{\kappa}(x,k\left( \omega\right) ) </math>.
As mentioned previously, determining <math>\zeta_\kappa</math> is the major
computation of the generalized eigenfunction method, but we simply
assume that they are known. We know that the eigenfunctions are
orthogonal for different <math>\omega</math> (from the self-adjointness of
<math>\partial_n\mathbf{G}</math>), and that the waves incident from the left and right
with the same <math>\omega</math> are orthogonal from the identity
<center><math>
R_1 T_{-1}^{*} + R_{-1}^{*} T_{1}=0,
</math></center>
[[Mei 1983]].
It therefore follows that
<center><math>
\left\langle \left( {\zeta}_{\kappa}(x,k\left( \omega_{1}\right) )\right)
,{\zeta}_{\kappa^{\prime}}(x,k\left( \omega_{2}\right) )\right\rangle _{\mathcal{H}
}=\Lambda_{n}\left( \omega_{1}\right) \delta\left( \omega_{1}-\omega
_{2}\right) \delta_{\kappa\kappa^{\prime}},
</math></center>
but we need to determine the normalizing function
<math>\Lambda_{n}\left( \omega_{n}\right)</math>. This is achieved by using the result that the
eigenfunctions satisfy the same normalizing condition with and without
the scatterers present. This result, the proof of which is quite
technical, is well-known and has been shown for many different
situations. The original proof was for Schr\"odinger's equation and was
due to [[Povzner 1953]] and [[Ikebe 1960]]. A proof for the case of Helmholtz
equation was given by [[Wilcox 1975]]. Recently the proof was given
for water waves by [[Hazard and Lenoir 2002]] and [[Hazard and Loret 2007]].

Since the eigenfunctions satisfy the same normalizing condition with
and without the scatterers, we normalize with the scatterers absent.
This means that the eigenfunctions are simply the incident waves, and
the free surface <math>F</math> is the entire axis. This allows us to derive
<center><math>\begin{matrix}
\left\langle \left( {\zeta}_{\kappa}(x,k\left( \omega_{1}\right) )\right)
,{\zeta}_{\kappa^{\prime}}(x,k\left( \omega_{2}\right) )\right\rangle _{\mathcal{H}}
&= \int_{\mathbb{R}}\left( e^{\kappa\mathrm{i} k_{1}
x}\right) \left( e^{\kappa^{\prime}\mathrm{i} k_{2}x}\right)
^{*}\,\mathrm{d} x \\
& =2\pi \delta_{\kappa\kappa^{\prime}}\delta\left( k_{1}-k_{2}\right) \\
& =2\pi\delta_{\kappa\kappa^{\prime}}\delta\left( \omega_{1}-\omega_{2}\right)
\left. \frac{d\omega}{dk}\right\vert _{\omega=\omega_{1}}.
\end{matrix}</math></center>
This result allows us to calculate the time-dependent solution in the
eigenfunctions (or single-frequency solutions).

==Expansion in Eigenfunctions==

We expand the solution for the displacement in the time domain as
<center><math>
\zeta\left( x,t\right)
=\int_{\mathbb{R}^{+}} \sum_{\kappa\in\left\{ -1,1\right\}}
\left\{
f_{\kappa}\left( \omega\right)
\cos(\omega t)+
g_{\kappa}\left( \omega\right)
\frac{\sin(\omega t)}{\omega}\right\}
\zeta_{\kappa}(x,k) d\omega,
</math></center>
where <math>f_\kappa</math> and <math>g_\kappa</math> will be determined from the initial
conditions. Note that here, and in subsequent equations, we are
assuming that <math>k</math> is a function of <math>\omega</math> or that <math>\omega</math> is a
function of <math>k</math> as required. If we take the inner product with
respect to the eigenfunctions <math>\zeta_\kappa</math> we obtain
<center><math>
\left\langle
\zeta_0\left( x\right)
,\zeta_{\kappa}(x,k)\right\rangle _{\mathcal{H}}=2\pi f_{\kappa}\left(
\omega\right) \frac{d\omega}{dk},
</math></center>
and
<center><math>
\left\langle
\partial_t\zeta_0\left( x\right)
,\zeta_{\kappa}(x,k)\right\rangle _{\mathcal{H}}=2\pi g_{\kappa}\left(
\omega\right) \frac{d\omega}{dk}.
</math></center>
We can therefore write, changing the
variable of integration to <math>k</math>, as
<center><math>
\zeta\left( x,t\right)
=\frac{1}{2\pi}\int_{\mathbb{R}^{+}} \sum_{\kappa\in\left\{ -1,1\right\}}
\Big\{
\left\langle \zeta_0\left( x\right)
,\zeta_{\kappa}(x,k)\right\rangle _{\mathcal{H}}
\cos(\omega t)
+ \left\langle
\partial_t \zeta_0\left( x\right)
,\zeta_{\kappa}(x,k)\right\rangle _{\mathcal{H}}
\frac{\sin(\omega t)}{\omega}\Big\}
\zeta_{\kappa}(x,k) dk,
</math></center>

If we take the case when <math>\partial_t\zeta_0( x) =0</math> and
write the integral given by the inner product explicitly, we obtain
<center><math>
\zeta\left( x,t\right) =\int_{\mathbb{R}^{+}}\Big\{ \sum_{\kappa\in\left\{
-1,1\right\} }\left( \frac{1}{2\pi}\int_{F}\zeta_0\left( x^{\prime}\right)
\zeta_{\kappa}(x^{\prime},k) ^{*}\,\mathrm{d} x^{\prime}\right)
\zeta_{\kappa}(x,k)\Big\}\cos(\omega t)dk.
</math></center>

If the scattering structure supports a trapped mode at a particular frequency <math>\omega_{0}</math> then the expression for the free-surface elevation becomes
<center><math>
\zeta\left( x,t\right) =\int_{\mathbb{R}^{+}}\Big\{ \sum_{\kappa\in\left\{
-1,1\right\} }\left( \frac{1}{2\pi}\int_{F}\zeta_0\left( x^{\prime}\right)
\zeta_{\kappa}(x^{\prime},k) ^{*}\,\mathrm{d} x^{\prime}\right)
\zeta_{\kappa}(x,k)\Big\}\cos(\omega t)dk + \frac{\int_{F}\zeta_{0}(x^{\prime})\tilde\zeta(x^{\prime})^{*}}{\int_{F}\tilde\zeta(x^{\prime})\tilde\zeta(x^{\prime})}d\x^{\prime} \cos(\omega_{0} t)
</math></center>
where <math>\tilde\zeta(x^{\prime})</math> is the trapped mode free-surface elevation.

Generalized Eigenfunction Expansion for Water Waves for a Fixed Body

2009-08-17T01:33:36Z

Cfitzgerald: /* Expansion in Eigenfunctions */

== Introduction ==

The generalized eigenfunction method goes back to
the work of [[Povzner 1953]] and [[Ikebe 1960]]. The generalized eigenfunction
method has been applied to water-wave problems by
[[Friedman and Shinbrot 1967]], [[Hazard_Lenoir2002a | Hazard and Lenoir 2002]]
(for the case of a rigid body in water of infinite depth)
and [[Meylan 2002b| Meylan 2002 ]] (for
a thin plate on water of shallow draft). We will present here the theory for a rigid body
in water of finite depth.
rms of this solutions (which we call the generalised
eigenfunctions) because they solve for

The generalized eigenfunction method is based on an inner product in
which the evolution operator is self-adjoint. It follows from the
self-adjointness that we can expand the solution in the
eigenfunctions of the operator. These eigenfunctions
are nothing more than the single-frequency solutions. The main
difficulty is that the eigenfunctions are associated with a continuous
spectrum, and this requires that they be carefully normalized. Once
this is done, we can derive simple expressions which allow the
time-domain problem to be solved in terms of the single-frequency
solutions. The mathematical ideas are discussed in detail in
[[Hazard and Loret 2007]].

{{Equations for fixed bodies in the time domain}}

== Single frequency equations ==

The single frequency solution is based on the assumption that all
time-dependence is given by <math>\mathrm{e}^{\mathrm{i}\omega t}</math> and that the system is
excited by an incident wave. We can then write
<center><math>
\Phi\left( \mathbf{x},t\right) ={\Phi}_\kappa\left( \mathbf{x},\omega\right)
\mathrm{e}^{-\mathrm{i}\omega t},\ \ \ \mathrm{and} \ \ \ \zeta\left(
x,t\right) ={\zeta}_\kappa\left( x,\omega\right) \mathrm{e}^{-\mathrm{i}\omega
t},
</math></center>
where <math>\kappa=1</math> for waves excited by an incident wave from the left
and <math>-1</math> for waves incident from the right.
Under these assumptions, the equations become
<center><math>
\Delta{\Phi}_\kappa\left( \mathbf{x,}\omega\right) =0,\ \ \mathbf{x}\in
\Omega,
</math></center>
<center><math>
\partial_{n}{\Phi}_\kappa=0,\ \ \mathbf{x}\in\partial\Omega,
</math></center>
<center><math>
\partial_{n}{\Phi}_\kappa=0,\ \ z=-h,
</math></center>
<center><math>
-\mathrm{i}\omega{\zeta}_\kappa=\partial_{n}{\Phi}_\kappa,\ \ z=0,\ x\in F,
</math></center>
<center><math>
{\zeta}_\kappa = \mathrm{i}\omega{\Phi}_\kappa,\ \ z=0,\ x\in F.
</math></center>

We must also specify radiations conditions, which are given
by
<center><math>
{\Phi}_1=\frac{1}{\mathrm{i} \omega} \left(
\mathrm{e}^{\mathrm{i} kx} + R_1 e^{-\mathrm{i} k x} \right)\frac{\cosh k\left( z+h\right)
}{\cosh kh},
\,\,\,\mathrm{as}\,\,x\to-\infty,
</math></center>
<center><math>
{\Phi}_1=\frac{1}{\mathrm{i} \omega}
T_1\mathrm{e}^{\mathrm{i} kx}\frac{\cosh k\left( z+h\right) }{\cosh kh},
\,\,\,\mathrm{as}\,\,x\to\infty,
</math></center>
<center><math>
{\Phi}_{-1}=\frac{1}{\mathrm{i} \omega}
T_{-1}\mathrm{e}^{-\mathrm{i} kx}
\frac{\cosh k\left( z+h\right) }{\cosh kh},
\,\,\,\mathrm{as}\,\,x\to\infty,
</math></center>
<center><math>
{\Phi}_{-1}=\frac{1}{\mathrm{i} \omega}
\left(
\mathrm{e}^{-\mathrm{i} kx} + R_{-1} e^{\mathrm{i} k x} \right)\frac{\cosh k\left( z+h\right)
}{\cosh kh},
\,\,\,\mathrm{as}\,\,x\to-\infty.
</math></center>

Note that <math>R_{\kappa}</math> and <math>T_{\kappa}</math> are the reflection and
transmission coefficients respectively and that we have normalized so
that the amplitude (in displacement) is unity. The wavenumber <math>k</math> is
the positive real solution of the [[Dispersion Relation for a Free Surface|dispersion equation]]
<math>k\tanh kh=\omega^{2}</math>,
and we will consider both <math>k(\omega)</math> and <math>\omega(k)</math>
in what follows as required. The solution of the single-frequency
equation may be computationally challenging and for the generalized
eigenfunction expansion the major numerical work is to determine the
single-frequency solutions.

== Time domain calculations ==

The solution in the frequency domain can be used to construct the
solution in the time domain. This is well-known for the case of a
plane incident wave which is initially far from the body, and in this
case the solution can be calculated straightforwardly using the standard Fourier
transform. However, when we consider an initial displacement which is
non-zero around the bodies, we cannot express the time-dependent
solution in terms of the single frequency solutions by a standard Fourier
transform.

We begin with the equations in the time domain subject to the initial conditions given by
Denoting the potential at the surface by
<center><math>
\phi(x,t)=\left.\Phi\left(\mathbf{x},t\right)\right|_{z=0},
</math></center>
we introduce the operator <math>\mathbf{G}</math> which maps the surface
potential to the potential throughout the fluid domain. The operator
<math>\mathbf{G}\psi</math> is found by solving
<center><math>\begin{matrix}
\Delta\Psi\left( \mathbf{x}\right) & = 0,\ \ \mathbf{x}\in\Omega,\\
\partial_{n}\Psi & = 0,\ \ \mathbf{x}\in\partial\Omega,\\
\partial_{n}\Psi & = 0,\ \ z=-h,\\
\Psi & =\psi,\ \ z = 0,\ x\in F,
\end{matrix}</math></center>
and is defined by <math>\mathbf{G}\psi=\Psi.</math> The operator
<math>\partial_{n}\mathbf{G}</math>, which maps the surface potential to the normal
derivative of potential at the surface (called the
Dirichlet-to-Neumann map) is given by
<center><math>
\partial_{n}\mathbf{G}\psi=\left. \partial_{n}\Psi\right\vert _{z=0}.
</math></center>
Therefore the equations in the time domain can be written as
<center><math>
\partial_{t}^2 \zeta + \partial_{n}\mathbf{G} \zeta = 0.
</math></center>
where we can recover the potential using the operator <math>\mathbf{G}</math>.
The evolution operator <math>\partial_{n}\mathbf{G}</math> is
symmetric in the Hilbert space given by the following inner product
<center><math>
\left\langle \zeta,\eta \right\rangle _{\mathcal{H}}=
\int_{F}\zeta \eta ^{*}\,\mathrm{d} x,
</math></center>
where <math> *</math> denotes complex conjugate, and we assume that this
symmetry implies that the operator is self-adjoint. We can prove the
symmetry by using Green's second identity
<center><math>\begin{matrix}
\int_{F}\left( \partial_{n}\mathbf{G}\zeta \right) \left(
\eta\right) ^{*}\,\mathrm{d} x
&=\int_{F}\left( \partial_{n}\mathbf{G}\partial_t\phi \right) \left(
\partial_t\psi\right) ^{*}\,\mathrm{d} x \\
&=\int_{F}\left( \partial_t\phi \right) \left(
\partial_{n}\mathbf{G} \partial_t\psi\right) ^{*}\,\mathrm{d} x
\\
&=\int_{F}\left( \zeta \right) \left(
\partial_{n}\mathbf{G}\eta\right) ^{*}\,\mathrm{d} x,
\end{matrix}</math></center>
where <math>\phi</math> and <math>\psi</math> are the surface potentials associated with
<math>\zeta</math> and <math>\eta</math> respectively.

== Eigenfunctions of <math>\partial_{n}\mathbf{G}</math> ==

The eigenfunctions of <math> \partial_{n}\mathbf{G}</math> satisfy
<center><math>
\partial_{n}\mathbf{G} \zeta = \omega^2 \zeta.
</math></center>

To solve
for the eigenfunctions of <math>\partial_{n}\mathbf{G}</math> we need to solve
the frequency-domain equations, and the radian frequency <math>\omega</math> is exactly
the eigenvalue. To actually calculate the eigenfunctions of
<math>\partial_{n}\mathbf{G}</math> we need to specify the incident wave
potential, and for each frequency we have two eigenfunctions (waves
incident from the left and from the right). It is possible for there
to exist point spectra for this operator which correspond to the
existence of a [[Trapped Modes|trapped mode]] [[McIver 1996]]
and the presence of a trapped mode requires that
the generalized eigenfunction expansion we derive must be modified.

==Normalization of the Eigenfunctions==

The eigenfunctions of <math>\partial_{n}\mathbf{G}</math> (with eigenvalue
<math>\omega</math>) are denoted by <math> \zeta_{\kappa}(x,k\left( \omega\right) ) </math>.
As mentioned previously, determining <math>\zeta_\kappa</math> is the major
computation of the generalized eigenfunction method, but we simply
assume that they are known. We know that the eigenfunctions are
orthogonal for different <math>\omega</math> (from the self-adjointness of
<math>\partial_n\mathbf{G}</math>), and that the waves incident from the left and right
with the same <math>\omega</math> are orthogonal from the identity
<center><math>
R_1 T_{-1}^{*} + R_{-1}^{*} T_{1}=0,
</math></center>
[[Mei 1983]].
It therefore follows that
<center><math>
\left\langle \left( {\zeta}_{\kappa}(x,k\left( \omega_{1}\right) )\right)
,{\zeta}_{\kappa^{\prime}}(x,k\left( \omega_{2}\right) )\right\rangle _{\mathcal{H}
}=\Lambda_{n}\left( \omega_{1}\right) \delta\left( \omega_{1}-\omega
_{2}\right) \delta_{\kappa\kappa^{\prime}},
</math></center>
but we need to determine the normalizing function
<math>\Lambda_{n}\left( \omega_{n}\right)</math>. This is achieved by using the result that the
eigenfunctions satisfy the same normalizing condition with and without
the scatterers present. This result, the proof of which is quite
technical, is well-known and has been shown for many different
situations. The original proof was for Schr\"odinger's equation and was
due to [[Povzner 1953]] and [[Ikebe 1960]]. A proof for the case of Helmholtz
equation was given by [[Wilcox 1975]]. Recently the proof was given
for water waves by [[Hazard and Lenoir 2002]] and [[Hazard and Loret 2007]].

Since the eigenfunctions satisfy the same normalizing condition with
and without the scatterers, we normalize with the scatterers absent.
This means that the eigenfunctions are simply the incident waves, and
the free surface <math>F</math> is the entire axis. This allows us to derive
<center><math>\begin{matrix}
\left\langle \left( {\zeta}_{\kappa}(x,k\left( \omega_{1}\right) )\right)
,{\zeta}_{\kappa^{\prime}}(x,k\left( \omega_{2}\right) )\right\rangle _{\mathcal{H}}
&= \int_{\mathbb{R}}\left( e^{\kappa\mathrm{i} k_{1}
x}\right) \left( e^{\kappa^{\prime}\mathrm{i} k_{2}x}\right)
^{*}\,\mathrm{d} x \\
& =2\pi \delta_{\kappa\kappa^{\prime}}\delta\left( k_{1}-k_{2}\right) \\
& =2\pi\delta_{\kappa\kappa^{\prime}}\delta\left( \omega_{1}-\omega_{2}\right)
\left. \frac{d\omega}{dk}\right\vert _{\omega=\omega_{1}}.
\end{matrix}</math></center>
This result allows us to calculate the time-dependent solution in the
eigenfunctions (or single-frequency solutions).

==Expansion in Eigenfunctions==

We expand the solution for the displacement in the time domain as
<center><math>
\zeta\left( x,t\right)
=\int_{\mathbb{R}^{+}} \sum_{\kappa\in\left\{ -1,1\right\}}
\left\{
f_{\kappa}\left( \omega\right)
\cos(\omega t)+
g_{\kappa}\left( \omega\right)
\frac{\sin(\omega t)}{\omega}\right\}
\zeta_{\kappa}(x,k) d\omega,
</math></center>
where <math>f_\kappa</math> and <math>g_\kappa</math> will be determined from the initial
conditions. Note that here, and in subsequent equations, we are
assuming that <math>k</math> is a function of <math>\omega</math> or that <math>\omega</math> is a
function of <math>k</math> as required. If we take the inner product with
respect to the eigenfunctions <math>\zeta_\kappa</math> we obtain
<center><math>
\left\langle
\zeta_0\left( x\right)
,\zeta_{\kappa}(x,k)\right\rangle _{\mathcal{H}}=2\pi f_{\kappa}\left(
\omega\right) \frac{d\omega}{dk},
</math></center>
and
<center><math>
\left\langle
\partial_t\zeta_0\left( x\right)
,\zeta_{\kappa}(x,k)\right\rangle _{\mathcal{H}}=2\pi g_{\kappa}\left(
\omega\right) \frac{d\omega}{dk}.
</math></center>
We can therefore write, changing the
variable of integration to <math>k</math>, as
<center><math>
\zeta\left( x,t\right)
=\frac{1}{2\pi}\int_{\mathbb{R}^{+}} \sum_{\kappa\in\left\{ -1,1\right\}}
\Big\{
\left\langle \zeta_0\left( x\right)
,\zeta_{\kappa}(x,k)\right\rangle _{\mathcal{H}}
\cos(\omega t)
+ \left\langle
\partial_t \zeta_0\left( x\right)
,\zeta_{\kappa}(x,k)\right\rangle _{\mathcal{H}}
\frac{\sin(\omega t)}{\omega}\Big\}
\zeta_{\kappa}(x,k) dk,
</math></center>

If we take the case when <math>\partial_t\zeta_0( x) =0</math> and
write the integral given by the inner product explicitly, we obtain
<center><math>
\zeta\left( x,t\right) =\int_{\mathbb{R}^{+}}\Big\{ \sum_{\kappa\in\left\{
-1,1\right\} }\left( \frac{1}{2\pi}\int_{F}\zeta_0\left( x^{\prime}\right)
\zeta_{\kappa}(x^{\prime},k) ^{*}\,\mathrm{d} x^{\prime}\right)
\zeta_{\kappa}(x,k)\Big\}\cos(\omega t)dk.
</math></center>

If the scattering structure supports a trapped mode at a particular frequency <math>\omega_{0}</math> then the expression for the free-surface elevation becomes
<center><math>
\zeta\left( x,t\right) =\int_{\mathbb{R}^{+}}\Big\{ \sum_{\kappa\in\left\{
-1,1\right\} }\left( \frac{1}{2\pi}\int_{F}\zeta_0\left( x^{\prime}\right)
\zeta_{\kappa}(x^{\prime},k) ^{*}\,\mathrm{d} x^{\prime}\right)
\zeta_{\kappa}(x,k)\Big\}\cos(\omega t)dk + \frac{\int_{F}\zeta_{0}(x^{\prime})\tilde\zeta(x^{\prime})^{*}}{\int_{F}\tilde\zeta(x^{\prime})\tilde\zeta(x^{\prime})\cos(\omega_{0} t)}d\omega
</math></center>
where <math>\tilde\zeta(x^{\prime})<\math> is the trapped mode free-surface elevation.

==An identity linking waves from the left and right==

A consequence of the requirement that the displacement be real, if the
initial displacement and initial derivative of displacement is real, is
that
<center><math>
\sum_{\kappa\in\left\{
-1,1\right\}} \left\langle \zeta_0\left( x\right)
,\zeta_{\kappa}(x,k)\right\rangle _{\mathcal{H}} \zeta_{\kappa}(x,k),
</math></center>
must be purely real. This can only be true if
<center><math>
\Im \left\{ \zeta_{1}(x^\prime,k)^{*} \zeta_{1}(x,k) \right\}
=
- \Im \left\{ \zeta_{-1}(x^\prime,k)^{*} \zeta_{-1}(x,k) \right\},
\,\,\,x,x^{\prime} \in F.
</math></center>

[[Category:Time-Dependent Linear Water Waves]]