WikiWaves - User contributions [en]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T13:39:35Z

Pun Wong: /* Solution with Waves Incident at an Angle */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\in\mathbb{N}\cup\left\{0\right\}</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\in\mathbb{N}\cup\left\{0\right\}.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain:
<center><math>
\begin{align}
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}a_{m} \phi_{m}\left( z\right)
&=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z), \quad -h<z<-d \\

-k_{0}\phi_{0}\left( z\right) +\sum_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) &=
\begin{cases}
\quad \quad \quad 0,\quad -d<z<0 \\
-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi_{m}(z),\quad -h<z<-d
\end{cases}
\end{align}
</math></center>

For the first equation we multiply both sides by <math>\psi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>-d</math> to obtain:
<center><math>
B_{n0} + \sum^{\infty}_{m=0}a_{m}B_{nm} = b_{m}C_{m}\delta_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math></center>
and for the second equation we multiply both sides by <math>\phi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center><math>
-k_{0}A_{0}\delta_{0n} + k_{m}a_{m}A_{m} = -\sum^{\infty}_{m=0}\kappa_{m}b_{m}B_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math></center>

Solving the equations above will yield the coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations, we set the upper limit of <math>n</math> to be <math>N</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
B_{00} & B_{01} & \cdots & B_{0N} & -C_{0} & 0 & \cdots & 0 \\
B_{10} & B_{11} & \cdots & B_{1N} & 0 & -C_{1} & \cdots & \vdots \\
\vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & 0 \\
B_{N0} & B_{N1} & \cdots & B_{NN} & 0 & \cdots & 0 & -C_{N} \\
k_{0}A_{0} & 0 & \cdots & 0 & \kappa_{0}B_{00} & \kappa_{1}B_{10} & \cdots & \kappa_{N}B_{N0} \\
0 & k_{1}A_{1} & \cdots & \vdots & \kappa_{0}B_{01} & \kappa_{1}B_{11} & \cdots & \kappa_{N}B_{N1} \\
\vdots & \vdots & \ddots & 0 & \vdots & \vdots & \ddots & \vdots \\
0 & \cdots & 0 & k_{N}A_{N} & \kappa_{0}B_{0N} & \kappa_{1}B_{1N} & \cdots & \kappa_{N}B_{NN} \\
\end{bmatrix}
\begin{bmatrix}
a_{0} \\ a_{1} \\ \vdots \\ a_{N} \\ b_{0} \\ b_{1} \\ \vdots \\ b_{N} \\
\end{bmatrix}
=\begin{bmatrix}
-B_{00} \\ -B_{10} \\ \vdots \\ -B_{N0} \\ k_{0}A_{0} \\ 0 \\ \vdots \\ 0 \\
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center><math>
\begin{align}
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}a_{m} \phi_{m}\left( z\right)
&=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z), \quad -h<z<-d \\

-\hat{k}_{0}\phi_{0}\left( z\right) +\sum_{m=0}^{\infty} a_{m}\hat{k}_{m}\phi_{m}\left( z\right) &=
\begin{cases}
\quad \quad \quad 0,\quad -d<z<0 \\
-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}\psi_{m}(z),\quad -h<z<-d
\end{cases}
\end{align}
</math></center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T13:38:31Z

Pun Wong: /* Solution with Waves Incident at an Angle */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\in\mathbb{N}\cup\left\{0\right\}</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\in\mathbb{N}\cup\left\{0\right\}.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain:
<center><math>
\begin{align}
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}a_{m} \phi_{m}\left( z\right)
&=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z), \quad -h<z<-d \\

-k_{0}\phi_{0}\left( z\right) +\sum_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) &=
\begin{cases}
\quad \quad \quad 0,\quad -d<z<0 \\
-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi_{m}(z),\quad -h<z<-d
\end{cases}
\end{align}
</math></center>

For the first equation we multiply both sides by <math>\psi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>-d</math> to obtain:
<center><math>
B_{n0} + \sum^{\infty}_{m=0}a_{m}B_{nm} = b_{m}C_{m}\delta_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math></center>
and for the second equation we multiply both sides by <math>\phi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center><math>
-k_{0}A_{0}\delta_{0n} + k_{m}a_{m}A_{m} = -\sum^{\infty}_{m=0}\kappa_{m}b_{m}B_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math></center>

Solving the equations above will yield the coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations, we set the upper limit of <math>n</math> to be <math>N</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
B_{00} & B_{01} & \cdots & B_{0N} & -C_{0} & 0 & \cdots & 0 \\
B_{10} & B_{11} & \cdots & B_{1N} & 0 & -C_{1} & \cdots & \vdots \\
\vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & 0 \\
B_{N0} & B_{N1} & \cdots & B_{NN} & 0 & \cdots & 0 & -C_{N} \\
k_{0}A_{0} & 0 & \cdots & 0 & \kappa_{0}B_{00} & \kappa_{1}B_{10} & \cdots & \kappa_{N}B_{N0} \\
0 & k_{1}A_{1} & \cdots & \vdots & \kappa_{0}B_{01} & \kappa_{1}B_{11} & \cdots & \kappa_{N}B_{N1} \\
\vdots & \vdots & \ddots & 0 & \vdots & \vdots & \ddots & \vdots \\
0 & \cdots & 0 & k_{N}A_{N} & \kappa_{0}B_{0N} & \kappa_{1}B_{1N} & \cdots & \kappa_{N}B_{NN} \\
\end{bmatrix}
\begin{bmatrix}
a_{0} \\ a_{1} \\ \vdots \\ a_{N} \\ b_{0} \\ b_{1} \\ \vdots \\ b_{N} \\
\end{bmatrix}
=\begin{bmatrix}
-B_{00} \\ -B_{10} \\ \vdots \\ -B_{N0} \\ k_{0}A_{0} \\ 0 \\ \vdots \\ 0 \\
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center><math>
\begin{align}
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}a_{m} \phi_{m}\left( z\right)
&=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z), \quad -h<z<-d \\

-\hat{k}_{0}\phi_{0}\left( z\right) +\sum_{m=0}^{\infty} a_{m}\hat{k}_{m}\phi_{m}\left( z\right) &=
\begin{cases}
\quad \quad \quad 0,\quad -d<z<0 \\
-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}\psi_{m}(z),\quad -h<z<-d
\end{cases}
\end{align}
</math></center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T13:32:21Z

Pun Wong: /* Numerical Solution */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\in\mathbb{N}\cup\left\{0\right\}</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\in\mathbb{N}\cup\left\{0\right\}.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain:
<center><math>
\begin{align}
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}a_{m} \phi_{m}\left( z\right)
&=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z), \quad -h<z<-d \\

-k_{0}\phi_{0}\left( z\right) +\sum_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) &=
\begin{cases}
\quad \quad \quad 0,\quad -d<z<0 \\
-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi_{m}(z),\quad -h<z<-d
\end{cases}
\end{align}
</math></center>

For the first equation we multiply both sides by <math>\psi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>-d</math> to obtain:
<center><math>
B_{n0} + \sum^{\infty}_{m=0}a_{m}B_{nm} = b_{m}C_{m}\delta_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math></center>
and for the second equation we multiply both sides by <math>\phi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center><math>
-k_{0}A_{0}\delta_{0n} + k_{m}a_{m}A_{m} = -\sum^{\infty}_{m=0}\kappa_{m}b_{m}B_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math></center>

Solving the equations above will yield the coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations, we set the upper limit of <math>n</math> to be <math>N</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
B_{00} & B_{01} & \cdots & B_{0N} & -C_{0} & 0 & \cdots & 0 \\
B_{10} & B_{11} & \cdots & B_{1N} & 0 & -C_{1} & \cdots & \vdots \\
\vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & 0 \\
B_{N0} & B_{N1} & \cdots & B_{NN} & 0 & \cdots & 0 & -C_{N} \\
k_{0}A_{0} & 0 & \cdots & 0 & \kappa_{0}B_{00} & \kappa_{1}B_{10} & \cdots & \kappa_{N}B_{N0} \\
0 & k_{1}A_{1} & \cdots & \vdots & \kappa_{0}B_{01} & \kappa_{1}B_{11} & \cdots & \kappa_{N}B_{N1} \\
\vdots & \vdots & \ddots & 0 & \vdots & \vdots & \ddots & \vdots \\
0 & \cdots & 0 & k_{N}A_{N} & \kappa_{0}B_{0N} & \kappa_{1}B_{1N} & \cdots & \kappa_{N}B_{NN} \\
\end{bmatrix}
\begin{bmatrix}
a_{0} \\ a_{1} \\ \vdots \\ a_{N} \\ b_{0} \\ b_{1} \\ \vdots \\ b_{N} \\
\end{bmatrix}
=\begin{bmatrix}
-B_{00} \\ -B_{10} \\ \vdots \\ -B_{N0} \\ k_{0}A_{0} \\ 0 \\ \vdots \\ 0 \\
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T13:31:46Z

Pun Wong: /* Numerical Solution */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\in\mathbb{N}\cup\left\{0\right\}</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\in\mathbb{N}\cup\left\{0\right\}.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain:
<center><math>
\begin{align}
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}a_{m} \phi_{m}\left( z\right)
&=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z), \quad -h<z<-d \\

-k_{0}\phi_{0}\left( z\right) +\sum_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) &=
\begin{cases}
\quad \quad \quad 0,\quad -d<z<0 \\
-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi_{m}(z),\quad -h<z<-d
\end{cases}
\end{align}
</math></center>

For the first equation we multiply both sides by <math>\psi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>-d</math> to obtain:
<center><math>
B_{n0} + \sum^{\infty}_{m=0}a_{m}B_{nm} = b_{m}C_{m}\delta_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math></center>
and for the second equation we multiply both sides by <math>\phi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center><math>
-k_{0}A_{0}\delta_{0n} + k_{m}a_{m}A_{m} = -\sum^{\infty}_{m=0}\kappa_{m}b_{m}B_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math></center>

Solving the equations above will yield the coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations, we set the upper limit of <math>n</math> to be <math>N</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
B_{00} & B_{01} & \cdots & B_{0N} & -C_{00} & 0 & \cdots & 0 \\
B_{10} & B_{11} & \cdots & B_{1N} & 0 & -C_{11} & \cdots & \vdots \\
\vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & 0 \\
B_{N0} & B_{N1} & \cdots & B_{NN} & 0 & \cdots & 0 & -C_{NN} \\
k_{0}A_{00} & 0 & \cdots & 0 & \kappa_{0}B_{00} & \kappa_{1}B_{10} & \cdots & \kappa_{N}B_{N0} \\
0 & k_{1}A_{11} & \cdots & \vdots & \kappa_{0}B_{01} & \kappa_{1}B_{11} & \cdots & \kappa_{N}B_{N1} \\
\vdots & \vdots & \ddots & 0 & \vdots & \vdots & \ddots & \vdots \\
0 & \cdots & 0 & k_{N}A_{NN} & \kappa_{0}B_{0N} & \kappa_{1}B_{1N} & \cdots & \kappa_{N}B_{NN} \\
\end{bmatrix}
\begin{bmatrix}
a_{0} \\ a_{1} \\ \vdots \\ a_{N} \\ b_{0} \\ b_{1} \\ \vdots \\ b_{N} \\
\end{bmatrix}
=\begin{bmatrix}
-B_{00} \\ -B_{10} \\ \vdots \\ -B_{N0} \\ k_{0}A_{00} \\ 0 \\ \vdots \\ 0 \\
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T13:29:27Z

Pun Wong: /* An infinite dimensional system of equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\in\mathbb{N}\cup\left\{0\right\}</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\in\mathbb{N}\cup\left\{0\right\}.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain:
<center><math>
\begin{align}
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}a_{m} \phi_{m}\left( z\right)
&=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z), \quad -h<z<-d \\

-k_{0}\phi_{0}\left( z\right) +\sum_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) &=
\begin{cases}
\quad \quad \quad 0,\quad -d<z<0 \\
-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi_{m}(z),\quad -h<z<-d
\end{cases}
\end{align}
</math></center>

For the first equation we multiply both sides by <math>\psi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>-d</math> to obtain:
<center><math>
B_{n0} + \sum^{\infty}_{m=0}a_{m}B_{nm} = b_{m}C_{m}\delta_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math></center>
and for the second equation we multiply both sides by <math>\phi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center><math>
-k_{0}A_{0}\delta_{0n} + k_{m}a_{m}A_{m} = -\sum^{\infty}_{m=0}\kappa_{m}b_{m}B_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math></center>

Solving the equations above will yield the coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T13:28:24Z

Pun Wong: /* An infinite dimensional system of equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\in\mathbb{N}\cup\left\{0\right\}</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\in\mathbb{N}\cup\left\{0\right\}.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain:
<center><math>
\begin{align}
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}a_{m} \phi_{m}\left( z\right)
&=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z), \quad -h<z<-d \\

-k_{0}\phi_{0}\left( z\right) +\sum_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) &=
\begin{cases}
\quad \quad \quad 0,\quad -d<z<0 \\
-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi_{m}(z),\quad -h<z<-d
\end{cases}
\end{align}
</math></center>

For the first equation we multiply both sides by <math>\psi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>-d</math> to obtain:
<center><math>
B_{n0} + \sum^{\infty}_{m=0}a_{m}B_{nm} = b_{m}C_{m}\delta_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math></center>
and for the second equation we multiply both sides by <math>\phi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
-k_{0}A_{0}\delta_{0n} + k_{m}a_{m}A_{m} = -\sum^{\infty}_{m=0}\kappa_{m}b_{m}B_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math>

Solving the equations above will yield the coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T13:26:49Z

Pun Wong: /* An infinite dimensional system of equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\in\mathbb{N}\cup\left\{0\right\}</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\in\mathbb{N}\cup\left\{0\right\}.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain:
<center><math>
\begin{align}
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}a_{m} \phi_{m}\left( z\right)
&=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z), \quad -h<z<-d \\

-k_{0}\phi_{0}\left( z\right) +\sum_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) &=
\begin{cases}
\quad \quad \quad 0,\quad -d<z<0 \\
-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi_{m}(z),\quad -h<z<-d
\end{cases}
\end{align}
</math></center>

For the first equation we multiply both sides by <math>\psi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>-d</math> to obtain:
<center><math>
B_{n0} + \sum^{\infty}_{m=0}a_{m}B_{nm} = b_{m}C_{m}\delta_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math></center>
and for the second equation we multiply both sides by <math>\phi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
-k_{0}A_{0}\delta_{0n} + k_{m}a_{m}A_{m} = -\sum^{\infty}_{m=0}\kappa_{m}b_{m}B_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math>

This equation gives the required equations to solve for the coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T13:26:29Z

Pun Wong: /* An infinite dimensional system of equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\in\mathbb{N}\cup\left\{0\right\}</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\in\mathbb{N}\cup\left\{0\right\}.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain:
<center><math>
\begin{align}
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}a_{m} \phi_{m}\left( z\right)
&=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z), \quad -h<z<-d

-k_{0}\phi_{0}\left( z\right) +\sum_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) &=
\begin{cases}
\quad \quad \quad 0,\quad -d<z<0 \\
-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi_{m}(z),\quad -h<z<-d
\end{cases}
</math></center>

For the first equation we multiply both sides by <math>\psi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>-d</math> to obtain:
<center><math>
B_{n0} + \sum^{\infty}_{m=0}a_{m}B_{nm} = b_{m}C_{m}\delta_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math></center>
and for the second equation we multiply both sides by <math>\phi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
-k_{0}A_{0}\delta_{0n} + k_{m}a_{m}A_{m} = -\sum^{\infty}_{m=0}\kappa_{m}b_{m}B_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math>

This equation gives the required equations to solve for the coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T13:25:12Z

Pun Wong: /* An infinite dimensional system of equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\in\mathbb{N}\cup\left\{0\right\}</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\in\mathbb{N}\cup\left\{0\right\}.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain:
<center><math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z), \quad -h<z<-d
</math></center>

<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) =
\begin{cases}
\quad \quad \quad 0,\quad -d<z<0 \\
-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi_{m}(z),\quad -h<z<-d
\end{cases}
</math></center>

For the first equation we multiply both sides by <math>\psi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>-d</math> to obtain:
<center><math>
B_{n0} + \sum^{\infty}_{m=0}a_{m}B_{nm} = b_{m}C_{m}\delta_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math></center>
and for the second equation we multiply both sides by <math>\phi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
-k_{0}A_{0}\delta_{0n} + k_{m}a_{m}A_{m} = -\sum^{\infty}_{m=0}\kappa_{m}b_{m}B_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math>

This equation gives the required equations to solve for the coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T13:24:49Z

Pun Wong: /* An infinite dimensional system of equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\in\mathbb{N}\cup\left\{0\right\}</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\in\mathbb{N}\cup\left\{0\right\}.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain:
<center><math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z), \quad -h<z<-d \\
</math></center>

<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) =
\begin{cases}
\quad \quad \quad 0,\quad -d<z<0 \\
-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi_{m}(z),\quad -h<z<-d
\end{cases}
</math></center>

For the first equation we multiply both sides by <math>\psi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>-d</math> to obtain:
<center><math>
B_{n0} + \sum^{\infty}_{m=0}a_{m}B_{nm} = b_{m}C_{m}\delta_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math></center>
and for the second equation we multiply both sides by <math>\phi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
-k_{0}A_{0}\delta_{0n} + k_{m}a_{m}A_{m} = -\sum^{\infty}_{m=0}\kappa_{m}b_{m}B_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math>

This equation gives the required equations to solve for the coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T13:24:40Z

Pun Wong: /* An infinite dimensional system of equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\in\mathbb{N}\cup\left\{0\right\}</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\in\mathbb{N}\cup\left\{0\right\}.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain:
<center><math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z), \quad -h<z<-d
</math></center>

<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) =
\begin{cases}
\quad \quad \quad 0,\quad -d<z<0 \\
-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi_{m}(z),\quad -h<z<-d
\end{cases}
</math></center>

For the first equation we multiply both sides by <math>\psi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>-d</math> to obtain:
<center><math>
B_{n0} + \sum^{\infty}_{m=0}a_{m}B_{nm} = b_{m}C_{m}\delta_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math></center>
and for the second equation we multiply both sides by <math>\phi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
-k_{0}A_{0}\delta_{0n} + k_{m}a_{m}A_{m} = -\sum^{\infty}_{m=0}\kappa_{m}b_{m}B_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math>

This equation gives the required equations to solve for the coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T13:24:21Z

Pun Wong: /* An infinite dimensional system of equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\in\mathbb{N}\cup\left\{0\right\}</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\in\mathbb{N}\cup\left\{0\right\}.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain:
<center><math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z), \quad -h<z<-d
</math></center>
<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) =
\begin{cases}
\quad \quad \quad 0,\quad -d<z<0 \\
-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi_{m}(z),\quad -h<z<-d
\end{cases}
</math></center>

For the first equation we multiply both sides by <math>\psi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>-d</math> to obtain:
<center><math>
B_{n0} + \sum^{\infty}_{m=0}a_{m}B_{nm} = b_{m}C_{m}\delta_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math></center>
and for the second equation we multiply both sides by <math>\phi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
-k_{0}A_{0}\delta_{0n} + k_{m}a_{m}A_{m} = -\sum^{\infty}_{m=0}\kappa_{m}b_{m}B_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math>

This equation gives the required equations to solve for the coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T13:23:35Z

Pun Wong: /* An infinite dimensional system of equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\in\mathbb{N}\cup\left\{0\right\}</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\in\mathbb{N}\cup\left\{0\right\}.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain:
<center><math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z), \quad -h<z<-d
</math></center>
, for <math>-d<z<0</math>:
<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) = 0
</math></center>
and for <math>-h<z<-d</math>:
<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math></center>

<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) =
\begin{cases}
0,\quad -d<z<0 \\
-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi_{m}(z),\quad -h<z<-d
\end{cases}
</math>
</center>

For the first equation we multiply both sides by <math>\psi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>-d</math> to obtain:
<center><math>
B_{n0} + \sum^{\infty}_{m=0}a_{m}B_{nm} = b_{m}C_{m}\delta_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math></center>
and for the second equation we multiply both sides by <math>\phi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
-k_{0}A_{0}\delta_{0n} + k_{m}a_{m}A_{m} = -\sum^{\infty}_{m=0}\kappa_{m}b_{m}B_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math>

This equation gives the required equations to solve for the coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T13:22:54Z

Pun Wong: /* An infinite dimensional system of equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\in\mathbb{N}\cup\left\{0\right\}</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\in\mathbb{N}\cup\left\{0\right\}.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain:
<center><math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z), -h<z<-d
</math></center>
, for <math>-d<z<0</math>:
<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) = 0
</math></center>
and for <math>-h<z<-d</math>:
<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math></center>

<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) =
\begin{cases}
0,\quad -d<z<0 \\
-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi_{m}(z),\quad -h<z<-d<
\end{cases}
</math>
</center>

For the first equation we multiply both sides by <math>\psi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>-d</math> to obtain:
<center><math>
B_{n0} + \sum^{\infty}_{m=0}a_{m}B_{nm} = b_{m}C_{m}\delta_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math></center>
and for the second equation we multiply both sides by <math>\phi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
-k_{0}A_{0}\delta_{0n} + k_{m}a_{m}A_{m} = -\sum^{\infty}_{m=0}\kappa_{m}b_{m}B_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math>

This equation gives the required equations to solve for the coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T13:22:31Z

Pun Wong: /* An infinite dimensional system of equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\in\mathbb{N}\cup\left\{0\right\}</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\in\mathbb{N}\cup\left\{0\right\}.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain:
<center><math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z), <math>-h<z<-d</math>
</math></center>
, for <math>-d<z<0</math>:
<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) = 0
</math></center>
and for <math>-h<z<-d</math>:
<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math></center>

<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) =
\begin{cases}
0,\quad -d<z<0 \\
-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi_{m}(z),\quad -h<z<-d<
\end{cases}
</math>
</center>

For the first equation we multiply both sides by <math>\psi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>-d</math> to obtain:
<center><math>
B_{n0} + \sum^{\infty}_{m=0}a_{m}B_{nm} = b_{m}C_{m}\delta_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math></center>
and for the second equation we multiply both sides by <math>\phi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
-k_{0}A_{0}\delta_{0n} + k_{m}a_{m}A_{m} = -\sum^{\infty}_{m=0}\kappa_{m}b_{m}B_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math>

This equation gives the required equations to solve for the coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T13:18:01Z

Pun Wong: /* An infinite dimensional system of equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\in\mathbb{N}\cup\left\{0\right\}</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\in\mathbb{N}\cup\left\{0\right\}.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain for <math>-h<z<-d</math>:
<center><math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math></center>
, for <math>-d<z<0</math>:
<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) = 0
</math></center>
and for <math>-h<z<-d</math>:
<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math></center>

For the first equation we multiply both sides by <math>\psi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>-d</math> to obtain:
<center><math>
B_{n0} + \sum^{\infty}_{m=0}a_{m}B_{nm} = b_{m}C_{m}\delta_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math></center>
and for the second equation we multiply both sides by <math>\phi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
-k_{0}A_{0}\delta_{0n} + k_{m}a_{m}A_{m} = -\sum^{\infty}_{m=0}\kappa_{m}b_{m}B_{mn}, n\in\mathbb{N}\cup\left\{0\right\}
</math>

This equation gives the required equations to solve for the coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T13:16:46Z

Pun Wong: /* An infinite dimensional system of equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\in\mathbb{N}\cup\left\{0\right\}</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\in\mathbb{N}\cup\left\{0\right\}.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain for <math>-h<z<-d</math>:
<center><math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math></center>
, for <math>-d<z<0</math>:
<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) = 0
</math></center>
and for <math>-h<z<-d</math>:
<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math></center>

For the first equation we multiply both sides by <math>\psi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>-d</math> to obtain:
<center><math>
B_{n0} + \sum^{\infty}_{m=0}a_{m}B_{nm} = b_{m}C_{m}\delta_{mn}, <math>n\in\mathbb{N}\cup\left\{0\right\}</math>
</math></center>
and for the second equation we multiply both sides by <math>\phi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
-k_{0}A_{00} + k_{m}a_{m}A_{m} = -\sum^{\infty}_{m=0}\kappa_{m}b_{m}B_{mn}
</math>

This equation gives the required equations to solve for the coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T13:16:32Z

Pun Wong: /* An infinite dimensional system of equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\in\mathbb{N}\cup\left\{0\right\}</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\in\mathbb{N}\cup\left\{0\right\}.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain for <math>-h<z<-d</math>:
<center><math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math></center>
, for <math>-d<z<0</math>:
<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) = 0
</math></center>
and for <math>-h<z<-d</math>:
<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math></center>

For the first equation we multiply both sides by <math>\psi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>-d</math> to obtain:
<center><math>
B_{n0} + \sum^{\infty}_{m=0}a_{m}B_{nm} = b_{m}C_{m}\delta_{mn}, \for <math>n\in\mathbb{N}\cup\left\{0\right\}</math>
</math></center>
and for the second equation we multiply both sides by <math>\phi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
-k_{0}A_{00} + k_{m}a_{m}A_{m} = -\sum^{\infty}_{m=0}\kappa_{m}b_{m}B_{mn}
</math>

This equation gives the required equations to solve for the coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T13:15:44Z

Pun Wong: /* An infinite dimensional system of equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\in\mathbb{N}\cup\left\{0\right\}</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\in\mathbb{N}\cup\left\{0\right\}.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain for <math>-h<z<-d</math>:
<center><math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math></center>
, for <math>-d<z<0</math>:
<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) = 0
</math></center>
and for <math>-h<z<-d</math>:
<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math></center>

For the first equation we multiply both sides by <math>\psi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>-d</math> to obtain: For <math>n\in\mathbb{N}\cup\left\{0\right\}</math>
<center><math>
B_{n0} + \sum^{\infty}_{m=0}a_{m}B_{nm} = b_{m}C_{m}\delta_{mn}
</math></center>
and for the second equation we multiply both sides by <math>\phi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
-k_{0}A_{00} + k_{m}a_{m}A_{m} = -\sum^{\infty}_{m=0}\kappa_{m}b_{m}B_{mn}
</math>

This equation gives the required equations to solve for the coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T13:15:10Z

Pun Wong: /* An infinite dimensional system of equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\in\mathbb{N}\cup\left\{0\right\}</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\in\mathbb{N}\cup\left\{0\right\}.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain for <math>-h<z<-d</math>:
<center><math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math></center>
, for <math>-d<z<0</math>:
<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) = 0
</math></center>
and for <math>-h<z<-d</math>:
<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math></center>

For the first equation we multiply both sides by <math>\psi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>-d</math> to obtain: For <math>n\in\mathbb{N}\cup\left\{0\right\}</math>
<center><math>
B_{n0}
</math></center>
and for the second equation we multiply both sides by <math>\phi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
-k_{0}A_{00} + k_{m}a_{m}A_{m} &= -\sum^{\infty}_{m=0}\kappa_{m}b_{m}B_{mn}
</math>

This equation gives the required equations to solve for the coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T13:14:12Z

Pun Wong: /* Separation of Variables for a Dock */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\in\mathbb{N}\cup\left\{0\right\}</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\in\mathbb{N}\cup\left\{0\right\}.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain for <math>-h<z<-d</math>:
<center><math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math></center>
, for <math>-d<z<0</math>:
<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) = 0
</math></center>
and for <math>-h<z<-d</math>:
<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math></center>

For the first equation we multiply both sides by <math>\psi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>-d</math> to obtain: For <math>n\in\mathbb{N}\cup\left\{0\right\}</math>
<center><math>
B_{n0} + \sum^{\infty}_{m=0}a_{m}B_{nm} &= b_{m}C_{m}\delta_{mn}
</math></center>
and for the second equation we multiply both sides by <math>\phi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
-k_{0}A_{00} + k_{m}a_{m}A_{m} &= -\sum^{\infty}_{m=0}\kappa_{m}b_{m}B_{mn}
</math>

This equation gives the required equations to solve for the coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T13:13:45Z

Pun Wong: /* Separation of Variables for a Dock */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\in\mathbb{N}\cup\left\{0\right\}</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\geq 0.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain for <math>-h<z<-d</math>:
<center><math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math></center>
, for <math>-d<z<0</math>:
<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) = 0
</math></center>
and for <math>-h<z<-d</math>:
<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math></center>

For the first equation we multiply both sides by <math>\psi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>-d</math> to obtain: For <math>n\in\mathbb{N}\cup\left\{0\right\}</math>
<center><math>
B_{n0} + \sum^{\infty}_{m=0}a_{m}B_{nm} &= b_{m}C_{m}\delta_{mn}
</math></center>
and for the second equation we multiply both sides by <math>\phi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
-k_{0}A_{00} + k_{m}a_{m}A_{m} &= -\sum^{\infty}_{m=0}\kappa_{m}b_{m}B_{mn}
</math>

This equation gives the required equations to solve for the coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T13:12:45Z

Pun Wong: /* An infinite dimensional system of equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\geq 0</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\geq 0.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain for <math>-h<z<-d</math>:
<center><math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math></center>
, for <math>-d<z<0</math>:
<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) = 0
</math></center>
and for <math>-h<z<-d</math>:
<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math></center>

For the first equation we multiply both sides by <math>\psi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>-d</math> to obtain: For <math>n\in\mathbb{N}\cup\left\{0\right\}</math>
<center><math>
B_{n0} + \sum^{\infty}_{m=0}a_{m}B_{nm} &= b_{m}C_{m}\delta_{mn}
</math></center>
and for the second equation we multiply both sides by <math>\phi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
-k_{0}A_{00} + k_{m}a_{m}A_{m} &= -\sum^{\infty}_{m=0}\kappa_{m}b_{m}B_{mn}
</math>

This equation gives the required equations to solve for the coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T12:54:27Z

Pun Wong: /* An infinite dimensional system of equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\geq 0</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\geq 0.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain for <math>-h<z<-d</math>:
<center><math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math></center>
, for <math>-d<z<0</math>:
<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) = 0
</math></center>
and for <math>-h<z<-d</math>:
<center><math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math></center>

We solve these equations by multiplying both equations by
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T12:49:58Z

Pun Wong: /* An infinite dimensional system of equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\geq 0</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\geq 0.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain for <math>-h<z<-d</math>
<center><math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math></center>
and

<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>m</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T12:48:11Z

Pun Wong: /* An infinite dimensional system of equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\geq 0</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\geq 0.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<math>For $-h<z<-d$</math>
<center><math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math></center>
and

<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>m</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T12:47:46Z

Pun Wong: /* An infinite dimensional system of equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\geq 0</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\geq 0.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<math>\text{For $-h<z<-d$}</math>
<center><math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math></center>
and

<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>m</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T12:38:47Z

Pun Wong: /* Inner product between free surface and dock modes */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\geq 0</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\geq 0.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math>
</center>
and
<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>m</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T12:36:33Z

Pun Wong: /* Inner product between free surface and dock modes */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\geq 0</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\geq 0.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}=\frac{\kappa_{m}\sin(\kappa_{m}h)\cos(k_{n}h)-k_{n}\cos(\kappa_{m}h)\sin(k_{n}h)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math>
</center>
and
<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>m</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T12:23:19Z

Pun Wong: /* Solution Method */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\geq 0</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\geq 0.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>

=== Inner product between free surface and dock modes ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z
= \int\nolimits_{-h}^{0} \frac{\cos(\kappa_{m}(z+h))\cos(k_{n}(z+h))}{\cos(k_{n}h)} \mathrm{d} z
</math></center>
where
<center><math>
B_{mn}=\frac{k_{n}\sin k_{n}h\cos\kappa_{m}h-\kappa_{m}\cos k_{n}h\sin
\kappa_{m}h}{\left( \cos k_{n}h\right) \left( k_{n}
^{2}-\kappa_{m}^{2}\right) }
</math></center>

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math>
</center>
and
<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>m</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T12:12:54Z

Pun Wong: /* Separation of Variables for a Dock */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\geq 0</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\geq 0.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2} \left( \frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}} \right)
</math>
</center>
{{free surface dock relations}}

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math>
</center>
and
<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>m</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T12:12:01Z

Pun Wong: /* Separation of Variables for a Dock */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\geq 0</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\geq 0.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2}\frac{\cos(\kappa_{m}(h-d))\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}}
</math>
</center>
{{free surface dock relations}}

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math>
</center>
and
<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>m</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T12:11:37Z

Pun Wong: /* Separation of Variables for a Dock */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\geq 0</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\geq 0.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} = \frac{1}{2}\frac{\cos(\kappa_{m}(h-d)\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}}
</math>
</center>
{{free surface dock relations}}

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math>
</center>
and
<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>m</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T12:10:30Z

Pun Wong: /* Separation of Variables for a Dock */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\geq 0</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\geq 0.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
where
<center>
<math>
C_{m} =
\begin{cases}
\frac{1}{2}\frac{\cos(\kappa_{m}(h-d)\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}},\quad m=n \\
0,\,\,\,m\neq n
\end{cases}
</math>
</center>
{{free surface dock relations}}

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math>
</center>
and
<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>m</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T12:09:44Z

Pun Wong: /* Separation of Variables for a Dock */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\geq 0</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\geq 0.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>
<center>
<math>
C_{m} =
\begin{cases}
\frac{1}{2}\frac{\cos(\kappa_{m}(h-d)\sin(\kappa_{m}(h-d))+\kappa_{m}(h-d)}{\kappa_{m}},\quad m=n \\
0,\,\,\,m\neq n
\end{cases}
</math>
</center>
{{free surface dock relations}}

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math>
</center>
and
<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>m</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T12:01:09Z

Pun Wong: /* Separation of Variables for a Dock */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\geq 0</math> and
<center><math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\geq 0.</math></center>
We note that
<center><math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math></center>

{{free surface dock relations}}

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math>
</center>
and
<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>m</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T11:51:49Z

Pun Wong: /* Solution Method */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for the fully submerged dock is given by:
<center><math>
Z^{\prime\prime} + k^2 Z =0,
</math></center>
subject to the boundary conditions
<center><math>
Z^{\prime} (-h) = 0,
</math></center>
and
<center><math>
Z^{\prime} (-d) = 0.
</math></center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\geq 0</math> and
<center>
<math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\geq 0.
</math>
</center>
We note that
<center>
<math>
\int\nolimits_{-h}^{0}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math>
</center>
where
<center>
<math>
C_{m} =
\begin{cases}
h,\quad m=0 \\
\frac{1}{2}h,\,\,\,m\neq 0
\end{cases}
</math>
</center>

{{free surface dock relations}}

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math>
</center>
and
<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>m</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T11:47:22Z

Pun Wong: /* Solution Method */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for a floating dockkkkkk
<center>
<math>
Z^{\prime\prime} + k^2 Z =0,
</math>
</center>
subject to the boundary conditions
<center>
<math>
Z^{\prime} (-h) = 0,
</math>
</center>
and
<center>
<math>
Z^{\prime} (0) = 0.
</math>
</center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h} \,</math>, <math>m\geq 0</math> and
<center>
<math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\geq 0.
</math>
</center>
We note that
<center>
<math>
\int\nolimits_{-h}^{0}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math>
</center>
where
<center>
<math>
C_{m} =
\begin{cases}
h,\quad m=0 \\
\frac{1}{2}h,\,\,\,m\neq 0
\end{cases}
</math>
</center>

{{free surface dock relations}}

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math>
</center>
and
<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>m</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T11:46:39Z

Pun Wong: /* Solution Method */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

=== Separation of Variables for a Dock ===

The separation of variables equation for a floating dock
<center>
<math>
Z^{\prime\prime} + k^2 Z =0,
</math>
</center>
subject to the boundary conditions
<center>
<math>
Z^{\prime} (-h) = 0,
</math>
</center>
and
<center>
<math>
Z^{\prime} (0) = 0.
</math>
</center>
The solution is
<math>k=\kappa_{m}= \frac{m\pi}{h} \,</math>, <math>m\geq 0</math> and
<center>
<math>
Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\geq 0.
</math>
</center>
We note that
<center>
<math>
\int\nolimits_{-h}^{0}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},
</math>
</center>
where
<center>
<math>
C_{m} =
\begin{cases}
h,\quad m=0 \\
\frac{1}{2}h,\,\,\,m\neq 0
\end{cases}
</math>
</center>

{{free surface dock relations}}

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math>
</center>
and
<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>m</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Template:Separation of variables for a free surface

2010-03-28T11:43:43Z

Pun Wong: /* Separation of variables for a free surface */

=== Separation of variables for a free surface ===

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables]

{{separation of variables in two dimensions}}

{{separation of variables for a free surface first part}}

We denote the
positive imaginary solution of this equation by <math>k_{0} \,</math> and
the positive real solutions by <math>k_{m} \,</math>, <math>m\geq1</math>. We define
<center>
<math>
\phi_{m}\left( z\right) =\frac{\cos k_{m}(z+h)}{\cos k_{m}h},\quad m\geq0
</math>
</center>
as the vertical eigenfunction of the potential in the open
water region. From [http://en.wikipedia.org/wiki/Sturm-Liouville_theory Sturm-Liouville theory] the
vertical eigenfunctions are orthogonal. They can be normalised to be orthonormal, but
this has no advantages for a numerical implementation. It can be shown that
<center>
<math>
\int\nolimits_{-h}^{0}\phi_{m}(z)\phi_{n}(z) \mathrm{d} z=A_{n}\delta_{mn}
</math>
</center>
where
<center>
<math>
A_{n}=\frac{1}{2}\left( \frac{\cos k_{n}h\sin k_{n}h+k_{n}h}{k_{n}\cos
^{2}k_{n}h}\right).
</math>
</center>

Template:Separation of variables for a free surface

2010-03-28T11:43:01Z

Pun Wong: /* Separation of variables for a free surface */

=== Separation of variables for a free surface ===

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables]

{{separation of variables in two dimensions}}

{{separation of variables for a free surface first part}}

We denote theee
positive imaginary solution of this equation by <math>k_{0} \,</math> and
the positive real solutions by <math>k_{m} \,</math>, <math>m\geq1</math>. We define
<center>
<math>
\phi_{m}\left( z\right) =\frac{\cos k_{m}(z+h)}{\cos k_{m}h},\quad m\geq0
</math>
</center>
as the vertical eigenfunction of the potential in the open
water region. From [http://en.wikipedia.org/wiki/Sturm-Liouville_theory Sturm-Liouville theory] the
vertical eigenfunctions are orthogonal. They can be normalised to be orthonormal, but
this has no advantages for a numerical implementation. It can be shown that
<center>
<math>
\int\nolimits_{-h}^{0}\phi_{m}(z)\phi_{n}(z) \mathrm{d} z=A_{n}\delta_{mn}
</math>
</center>
where
<center>
<math>
A_{n}=\frac{1}{2}\left( \frac{\cos k_{n}h\sin k_{n}h+k_{n}h}{k_{n}\cos
^{2}k_{n}h}\right).
</math>
</center>

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T11:37:05Z

Pun Wong: /* Governing Equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

{{separation of variables for a dock}}

{{free surface dock relations}}

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math>
</center>
and
<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>m</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T11:36:32Z

Pun Wong: /* Governing Equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>. \\
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

{{separation of variables for a dock}}

{{free surface dock relations}}

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math>
</center>
and
<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>m</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T11:36:22Z

Pun Wong: /* Governing Equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.
The boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

{{separation of variables for a dock}}

{{free surface dock relations}}

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math>
</center>
and
<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>m</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T11:35:33Z

Pun Wong: /* Governing Equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>. The
boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

{{separation of variables for a dock}}

{{free surface dock relations}}

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math>
</center>
and
<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>m</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T11:35:18Z

Pun Wong: /* Governing Equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>. The
boundary value problem can therefore be expressed as
<center><math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math></center>
<center><math>
\Delta\phi=0, \,\, -h<z<-d, \,\, x<0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,
</math></center>

We must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

{{separation of variables for a dock}}

{{free surface dock relations}}

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math>
</center>
and
<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>m</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T11:32:35Z

Pun Wong: /* Governing Equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>. The
boundary value problem can therefore be expressed as
<center>
<math>
\Delta\phi=0, \,\, -h<z<0, \,\, x<0
</math>
</center>
<center>
<math>
\partial_z\phi=0, \,\, z=-h,
</math>
</center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center>
<math>
\partial_z\phi=0, \,\, z=0,\,x>0,
</math>
</center>
We
must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

{{separation of variables for a dock}}

{{free surface dock relations}}

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math>
</center>
and
<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>m</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T11:31:52Z

Pun Wong: /* Governing Equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>. The
boundary value problem can therefore be expressed as
<center>
<math>
\Delta\phi=0, \,\, -h<z<0,
</math>
</center>
<center>
<math>
\partial_z\phi=0, \,\, z=-h,
</math>
</center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center>
<math>
\partial_z\phi=0, \,\, z=0,\,x>0,
</math>
</center>
We
must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

{{separation of variables for a dock}}

{{free surface dock relations}}

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math>
</center>
and
<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>m</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T11:31:35Z

Pun Wong: /* Governing Equations */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>. The
boundary value problem can therefore be expressed as
<center>
<math>
\Delta\phi=0, -h<z<0,
</math>
</center>
<center>
<math>
\partial_z\phi=0, \,\, z=-h,
</math>
</center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center>
<math>
\partial_z\phi=0, \,\, z=0,\,x>0,
</math>
</center>
We
must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

{{separation of variables for a dock}}

{{free surface dock relations}}

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math>
</center>
and
<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>m</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]

Eigenfunction Matching for a Semi-Infinite Rectangle

2010-03-28T11:22:43Z

Pun Wong: /* Introduction */

{{incomplete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> fully submerged through which not flow is possible.
We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional) and later we also consider the case when the waves are incident at an angle.

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== Governing Equations ==

We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
The depth of submergence is <math>d</math>.
The water is assumed to have constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>. The
boundary value problem can therefore be expressed as
<center>
<math>
\Delta\phi=0, \,\, -h<z<0,
</math>
</center>
<center>
<math>
\partial_z\phi=0, \,\, z=-h,
</math>
</center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center>
<math>
\partial_z\phi=0, \,\, z=0,\,x>0,
</math>
</center>
We
must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.

== Solution Method ==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

{{separation of variables for a dock}}

{{free surface dock relations}}

=== Expansion of the potential ===

We need to apply some boundary conditions at plus and minus infinity,
where are essentially the the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave.

Therefore the scattered potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== An infinite dimensional system of equations ===

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math>
</center>
and
<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>m</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
We then simply need to solve the linear system of equations.

== Solution with Waves Incident at an Angle ==

We can consider the problem when the waves are incident at an angle <math>\theta</math>.

{{incident angle}}

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\kappa}_{m} = \sqrt{\kappa_m^2 - k_y^2}</math>
where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml}
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]