Difference between revisions of "Eigenfunction Matching for a Semi-Infinite Rectangle"

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== Introduction ==
 
== 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.  
+
The problem consists of a region to the left with a free water surface and a region to the right with a rigid fixed semi infinite rectangular block with height <math>d + ...</math> freeboard with depth of submergence being <math>d</math> through which no 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.
+
We begin with the simple problem when the waves are normally incident (so that the problem is truly two-dimensional). We can extend this solution to a finite dock using symmetry
 
+
[[Eigenfunction Matching for a Finite Rectangle using Symmetry]]
[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]
 
  
 
== Governing Equations ==
 
== Governing Equations ==
  
We begin with the [[Frequency Domain Problem]] for a submerged dock which occupies
+
We begin with the [[Frequency Domain Problem]] for a partially submerged dock which occupies
 
the region <math>x>0</math> (we assume <math>e^{i\omega t}</math> time dependence).
 
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 depth of submergence is <math>d</math>.
Line 16: Line 15:
 
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>.  
 
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
 
The boundary value problem can therefore be expressed as
 +
 
<center><math>
 
<center><math>
\Delta\phi=0, \,\,  -h<z<0, \,\, x<0
+
\begin{align}
</math></center>
+
\Delta\phi &= 0, \,\,\,  -h\leq z\leq 0, \,\, x < 0, \\
<center><math>
+
 
\Delta\phi=0, \,\,  -h<z<-d, \,\, x>0
+
\Delta\phi &= 0, \,\,\,  -h\leq z\leq -d, \,\, x > 0. \\
</math></center>
+
 
<center><math>
+
\partial_z\phi &= \alpha\phi, \,\,\, z=0, \,\, x<0, \\
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
+
 
</math></center>
+
\partial_x\phi &= 0, \,\,\, -d\leq z\leq 0, \,\, x=0, \\
<center><math>
+
 
\partial_z\phi=0, \,\, z=-d,\,x>0,
+
\partial_z\phi &= 0, \,\,\, z=-d, \,\, x>0, \\
</math></center>
+
 
<center><math>
+
\partial_z\phi &= 0, \,\,\, z=-h.
\partial_z\phi=0, \,\, z=-h,
+
\end{align}
 
</math></center>
 
</math></center>
 +
 +
Where <math>\alpha = \omega^2/g</math>.
  
 
We must also apply the [[Sommerfeld Radiation Condition]]
 
We must also apply the [[Sommerfeld Radiation Condition]]
Line 39: Line 41:
 
== Solution Method ==
 
== Solution Method ==
  
We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
+
We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x\leq 0</math>
and <math>x>0</math>.
+
and <math>x\geq 0</math>.
  
 
{{separation of variables in two dimensions}}
 
{{separation of variables in two dimensions}}
Line 48: Line 50:
 
=== Separation of Variables for a Dock ===
 
=== Separation of Variables for a Dock ===
  
The separation of variables equation for the fully submerged dock is given by:
+
The separation of variables equation for a partially submerged dock is given by:
 
<center><math>
 
<center><math>
 
Z^{\prime\prime} + k^2 Z =0,
 
Z^{\prime\prime} + k^2 Z =0,
Line 61: Line 63:
 
</math></center>
 
</math></center>
 
The solution is  
 
The solution is  
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math>, <math>m\in\mathbb{N}\cup\left\{0\right\}</math> and
+
<math>k=\kappa_{m}= \frac{m\pi}{h-d} \,</math> , <math>m\in\mathbb{N}\cup\left\{0\right\}</math> and
 
<center><math>
 
<center><math>
 
Z = \psi_{m}\left(  z\right)  = \cos\kappa_{m}(z+h),\quad
 
Z = \psi_{m}\left(  z\right)  = \cos\kappa_{m}(z+h),\quad
Line 72: Line 74:
 
<center>
 
<center>
 
<math>
 
<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)
+
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>
 
</math>
 
</center>
 
</center>
 +
In addition there is the special case of when <math>m=n=0</math> ,
 +
<center>
 +
<math>
 +
C_{0} = h-d,
 +
</math>
 +
</center>
 +
where <math>\delta_{00}=1</math> .
  
 
=== Inner product between free surface and dock modes ===
 
=== Inner product between free surface and dock modes ===
  
 
<center><math>
 
<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\phi_{n}(z) \mathrm{d} z  
+
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\chi_{n}(z) \mathrm{d} z , \quad m\geq1, \quad n\geq0,
 
</math></center>
 
</math></center>
 
where
 
where
 
<center><math>
 
<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)}
+
B_{mn}= \int\nolimits_{-h}^{-d} \frac{\cos\kappa_{m}(z+h)\cos k_{n}(z+h)}{\cos(k_{n}h)} \mathrm{d} z=\frac{-(-1)^m k_{n}\sin k_{n}(h-d)}
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}
+
{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}.
 +
</math></center>
 +
The evaluation of this inner product works on the assumption that the two roots <math>\kappa_{m}</math> and <math>k_{n}</math> do not coincide. The Matlab code given below has a routine to check for this case and make necessary adjustments.
 +
 
 +
We also have the case where the functions of the inner product are switched.
 +
<center><math>
 +
B_{nm} = \int\nolimits_{-h}^{-d}\chi_{m}(z)\psi_{n}(z) \mathrm{d} z , \quad m\geq0, \quad n\geq0.
 +
</math></center>
 +
For <math>m \geq 0</math> and <math>n \geq 1</math> the evaluation is above with the exception that the indices of the expression are switched.
 +
 
 +
Finally there is the special case of when <math>m \geq 0</math> and <math>n = 0</math>.
 +
<center><math>
 +
B_{0m} = \frac{\sin k_{m}(h-d)}{k_{m}\cos(k_{m}h)}.
 
</math></center>
 
</math></center>
  
 
=== Expansion of the potential ===  
 
=== Expansion of the potential ===  
  
We need to apply some boundary conditions at plus and minus infinity,  
+
Before we can write down the expansion of the potentials we need to solve for the horizontal eigenfunctions. We start with the equation
where are essentially the the solution cannot grow. This means that we
+
<center>
only have the positive (or negative) roots of the dispersion equation.
+
<math>
However, it does not help us with the purely imaginary root. Here we
+
X^{\prime\prime} - k^2 X = 0,
must use a different condition, essentially identifying one solution
+
</math>
as the incoming wave and the other as the outgoing wave.  
+
</center>
 +
which has the general solution
 +
<center>
 +
<math>
 +
X(x) = a e^{kx} + b e^{-kx},
 +
</math>
 +
</center>
 +
where <math>a</math> and <math>b</math> are constants to be determined from the boundary conditions. The general solution, or eigenfunction expansion, for the potential <math>\phi</math> is obtained by superimposing all the possible modes; for the region of the '''free surface''' we have
 +
<center>
 +
<math>
 +
\phi(x,z)=\sum_{m=0}^{\infty}\left(a_{m}e^{k_{m}x} + b_{m}e^{-k_{m}x}\right)\chi_{m}(z), \quad x<0.
 +
</math>
 +
</center>
 +
In this expression the term involving <math>e^{ikx}</math> (recall that <math>k_{0} = -ik</math>) corresponds to a wave propagating towards large positive <math>x</math> while the term involving <math>e^{-ikx}</math> corresponds to a wave propagating towards large negative <math>x</math>. Terms which decay as <math>x \rightarrow -\infty</math> or as <math>x \rightarrow \infty</math> are referred to as evanescent modes.
 +
 
 +
The expansion of the potential for the region under the '''dock''' is
 +
<center>
 +
<math>
 +
\phi(x,z)=\sum_{m=0}^{\infty}\left(a_{m}e^{\kappa_{m}x} + b_{m}e^{-\kappa_{m}x}\right)\psi_{m}(z), \quad x>0.
 +
</math>
 +
</center>
 +
To obtain a unique solution, the diffracted field <math>\phi_{\mathrm{D}}</math> must satisfy the Sommerfeld radiation condition (see further above) specifying that the waves of this potential propagate away from the structure and the solution be bounded.
  
Therefore the scattered potential (without the incident wave, which will
+
Application of the radiation condition to both the general solutions immediately above give diffracted potentials of
be added later) can
 
be expanded as
 
 
<center>
 
<center>
 
<math>
 
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
+
\phi_{\mathrm{D}}(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\chi_{m}(z), \quad x<0
 
</math>
 
</math>
 
</center>
 
</center>
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<center>
 
<center>
 
<math>
 
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
+
\phi_{\mathrm{D}}(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
+
e^{-\kappa_{m}x}\psi_{m}(z), \quad x>0
 
</math>
 
</math>
 
</center>
 
</center>
 
where <math>a_{m}</math> and <math>b_{m}</math>
 
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
+
are the coefficients of the potential in the '''open water''' and
the dock covered region respectively.
+
the '''dock''' covered region respectively.
  
 
{{incident potential for two dimensions}}
 
{{incident potential for two dimensions}}
Line 121: Line 161:
  
 
The potential and its derivative must be continuous across the
 
The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
+
transition from open water to the dock covered region. Therefore, the
 
potentials and their derivatives at <math>x=0</math> have to be equal.
 
potentials and their derivatives at <math>x=0</math> have to be equal.
 
We obtain:
 
We obtain:
 
<center><math>
 
<center><math>
 
\begin{align}
 
\begin{align}
\phi_{0}\left(  z\right) + \sum_{m=0}^{\infty}a_{m} \phi_{m}\left(  z\right)  
+
\chi_{0}\left(  z\right) + \sum_{m=0}^{\infty}a_{m} \chi_{m}\left(  z\right)  
&=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z), \quad -h<z<-d \\
+
&=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z), \quad -h \leq z\leq -d \\
  
-k_{0}\phi_{0}\left(  z\right) +\sum_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left(  z\right) &=  
+
-k_{0}\chi_{0}\left(  z\right) +\sum_{m=0}^{\infty} a_{m}k_{m}\chi_{m}\left(  z\right) &=  
 
\begin{cases}
 
\begin{cases}
\quad \quad \quad 0,\quad -d<z<0 \\
+
\quad \quad \quad 0,\quad \quad \quad \quad -d\leq z\leq 0 \\
-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi_{m}(z),\quad -h<z<-d
+
-\sum_{m=1}^{\infty}b_{m}\kappa_{m}\psi_{m}(z),\quad -h\leq z\leq -d
 
\end{cases}
 
\end{cases}
 
\end{align}
 
\end{align}
Line 139: Line 179:
 
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 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>
 
<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\}
+
B_{n0} + \sum^{\infty}_{m=0}a_{m}B_{nm} = b_{m}C_{m}\delta_{mn}, \quad n\in\mathbb{N}\cup\left\{0\right\}
 
</math></center>
 
</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:
+
and for the second equation we multiply both sides by <math>\chi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
 
<center><math>
 
<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\}
+
  -k_{0}A_{0}\delta_{0n} + k_{m}a_{m}A_{m}\delta_{mn} = -\sum^{\infty}_{m=1}\kappa_{m}b_{m}B_{mn}, \quad n\in\mathbb{N}\cup\left\{0\right\}
 
</math></center>
 
</math></center>
  
Solving the equations above will yield the coefficients of the water velocity potential in the dock covered region.
+
Remember <math>\kappa_{0} = 0 \,</math>. Solving the equations above will yield the coefficients of the water velocity potential in the free surface and dock covered regions.
  
 
== Numerical Solution ==
 
== 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,
+
To solve the system of equations, we set the upper limit of <math>m</math> to be <math>N</math>. This resulting system can be expressed in the block matrix form below,
 +
 
 
<center>
 
<center>
 
<math>
 
<math>
Line 158: Line 199:
 
\vdots          & \vdots      & \ddots & \vdots      & \vdots          & \vdots          & \ddots    & 0        \\
 
\vdots          & \vdots      & \ddots & \vdots      & \vdots          & \vdots          & \ddots    & 0        \\
 
B_{N0}          & B_{N1}      & \cdots & B_{NN}      & 0                & \cdots          & 0        & -C_{N}  \\
 
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} \\
+
     k_{0}A_{0} & 0          & \cdots & 0           & 0               & \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} \\
+
         0        & k_{1}A_{1} & \cdots & \vdots       & 0               & \kappa_{1}B_{11} & \cdots    & \kappa_{N}B_{N1} \\
 
       \vdots    &  \vdots    & \ddots & 0          & \vdots          & \vdots          & \ddots    & \vdots          \\
 
       \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} \\
+
         0      &  \cdots    & 0      & k_{N}A_{N} & 0               & \kappa_{1}B_{1N} & \cdots    & \kappa_{N}B_{NN} \\
 
\end{bmatrix}
 
\end{bmatrix}
 
\begin{bmatrix}
 
\begin{bmatrix}
Line 171: Line 212:
 
</math>
 
</math>
 
</center>
 
</center>
 +
 
We then simply need to solve the linear system of equations.
 
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 ==
 
== Matlab Code ==
  
A program to calculate the coefficients for the semi-infinite dock problems can be found here
+
A program to calculate the coefficients for the semi-infinite dock problem can be found here
 
{{semiinfinite_dock code}}
 
{{semiinfinite_dock code}}
  

Latest revision as of 01:28, 16 March 2012


Introduction

The problem consists of a region to the left with a free water surface and a region to the right with a rigid fixed semi infinite rectangular block with height [math]\displaystyle{ d + ... }[/math] freeboard with depth of submergence being [math]\displaystyle{ d }[/math] through which no flow is possible. We begin with the simple problem when the waves are normally incident (so that the problem is truly two-dimensional). We can extend this solution to a finite dock using symmetry Eigenfunction Matching for a Finite Rectangle using Symmetry

Governing Equations

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

[math]\displaystyle{ \begin{align} \Delta\phi &= 0, \,\,\, -h\leq z\leq 0, \,\, x \lt 0, \\ \Delta\phi &= 0, \,\,\, -h\leq z\leq -d, \,\, x \gt 0. \\ \partial_z\phi &= \alpha\phi, \,\,\, z=0, \,\, x\lt 0, \\ \partial_x\phi &= 0, \,\,\, -d\leq z\leq 0, \,\, x=0, \\ \partial_z\phi &= 0, \,\,\, z=-d, \,\, x\gt 0, \\ \partial_z\phi &= 0, \,\,\, z=-h. \end{align} }[/math]

Where [math]\displaystyle{ \alpha = \omega^2/g }[/math].

We must also apply the Sommerfeld Radiation Condition as [math]\displaystyle{ |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 separation of variables in the two regions, [math]\displaystyle{ x\leq 0 }[/math] and [math]\displaystyle{ x\geq 0 }[/math].

We express the potential as

[math]\displaystyle{ \phi(x,z) = X(x)Z(z)\, }[/math]

and then Laplace's equation becomes

[math]\displaystyle{ \frac{X^{\prime\prime}}{X} = - \frac{Z^{\prime\prime}}{Z} = k^2 }[/math]

Separation of variables for a free surface

We use separation of variables

We express the potential as

[math]\displaystyle{ \phi(x,z) = X(x)Z(z)\, }[/math]

and then Laplace's equation becomes

[math]\displaystyle{ \frac{X^{\prime\prime}}{X} = - \frac{Z^{\prime\prime}}{Z} = k^2 }[/math]

The separation of variables equation for deriving free surface eigenfunctions is as follows:

[math]\displaystyle{ Z^{\prime\prime} + k^2 Z =0. }[/math]

subject to the boundary conditions

[math]\displaystyle{ Z^{\prime}(-h) = 0 }[/math]

and

[math]\displaystyle{ Z^{\prime}(0) = \alpha Z(0) }[/math]

We can then use the boundary condition at [math]\displaystyle{ z=-h \, }[/math] to write

[math]\displaystyle{ Z = \frac{\cos k(z+h)}{\cos kh} }[/math]

where we have chosen the value of the coefficent so we have unit value at [math]\displaystyle{ z=0 }[/math]. The boundary condition at the free surface ([math]\displaystyle{ z=0 \, }[/math]) gives rise to:

[math]\displaystyle{ k\tan\left( kh\right) =-\alpha \, }[/math]

which is the Dispersion Relation for a Free Surface

The above equation is a transcendental equation. If we solve for all roots in the complex plane we find that the first root is a pair of imaginary roots. We denote the imaginary solutions of this equation by [math]\displaystyle{ k_{0}=\pm ik \, }[/math] and the positive real solutions by [math]\displaystyle{ k_{m} \, }[/math], [math]\displaystyle{ m\geq1 }[/math]. The [math]\displaystyle{ k \, }[/math] of the imaginary solution is the wavenumber. We put the imaginary roots back into the equation above and use the hyperbolic relations

[math]\displaystyle{ \cos ix = \cosh x, \quad \sin ix = i\sinh x, }[/math]

to arrive at the dispersion relation

[math]\displaystyle{ \alpha = k\tanh kh. }[/math]

We note that for a specified frequency [math]\displaystyle{ \omega \, }[/math] the equation determines the wavenumber [math]\displaystyle{ k \, }[/math].

Finally we define the function [math]\displaystyle{ Z(z) \, }[/math] as

[math]\displaystyle{ \chi_{m}\left( z\right) =\frac{\cos k_{m}(z+h)}{\cos k_{m}h},\quad m\geq0 }[/math]

as the vertical eigenfunction of the potential in the open water region. From 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

[math]\displaystyle{ \int\nolimits_{-h}^{0}\chi_{m}(z)\chi_{n}(z) \mathrm{d} z=A_{n}\delta_{mn} }[/math]

where

[math]\displaystyle{ 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]

Separation of Variables for a Dock

The separation of variables equation for a partially submerged dock is given by:

[math]\displaystyle{ Z^{\prime\prime} + k^2 Z =0, }[/math]

subject to the boundary conditions

[math]\displaystyle{ Z^{\prime} (-h) = 0, }[/math]

and

[math]\displaystyle{ Z^{\prime} (-d) = 0. }[/math]

The solution is [math]\displaystyle{ k=\kappa_{m}= \frac{m\pi}{h-d} \, }[/math] , [math]\displaystyle{ m\in\mathbb{N}\cup\left\{0\right\} }[/math] and

[math]\displaystyle{ Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad m\in\mathbb{N}\cup\left\{0\right\}. }[/math]

We note that

[math]\displaystyle{ \int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn}, }[/math]

where

[math]\displaystyle{ 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]

In addition there is the special case of when [math]\displaystyle{ m=n=0 }[/math] ,

[math]\displaystyle{ C_{0} = h-d, }[/math]

where [math]\displaystyle{ \delta_{00}=1 }[/math] .

Inner product between free surface and dock modes

[math]\displaystyle{ B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\chi_{n}(z) \mathrm{d} z , \quad m\geq1, \quad n\geq0, }[/math]

where

[math]\displaystyle{ B_{mn}= \int\nolimits_{-h}^{-d} \frac{\cos\kappa_{m}(z+h)\cos k_{n}(z+h)}{\cos(k_{n}h)} \mathrm{d} z=\frac{-(-1)^m k_{n}\sin k_{n}(h-d)} {\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}. }[/math]

The evaluation of this inner product works on the assumption that the two roots [math]\displaystyle{ \kappa_{m} }[/math] and [math]\displaystyle{ k_{n} }[/math] do not coincide. The Matlab code given below has a routine to check for this case and make necessary adjustments.

We also have the case where the functions of the inner product are switched.

[math]\displaystyle{ B_{nm} = \int\nolimits_{-h}^{-d}\chi_{m}(z)\psi_{n}(z) \mathrm{d} z , \quad m\geq0, \quad n\geq0. }[/math]

For [math]\displaystyle{ m \geq 0 }[/math] and [math]\displaystyle{ n \geq 1 }[/math] the evaluation is above with the exception that the indices of the expression are switched.

Finally there is the special case of when [math]\displaystyle{ m \geq 0 }[/math] and [math]\displaystyle{ n = 0 }[/math].

[math]\displaystyle{ B_{0m} = \frac{\sin k_{m}(h-d)}{k_{m}\cos(k_{m}h)}. }[/math]

Expansion of the potential

Before we can write down the expansion of the potentials we need to solve for the horizontal eigenfunctions. We start with the equation

[math]\displaystyle{ X^{\prime\prime} - k^2 X = 0, }[/math]

which has the general solution

[math]\displaystyle{ X(x) = a e^{kx} + b e^{-kx}, }[/math]

where [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] are constants to be determined from the boundary conditions. The general solution, or eigenfunction expansion, for the potential [math]\displaystyle{ \phi }[/math] is obtained by superimposing all the possible modes; for the region of the free surface we have

[math]\displaystyle{ \phi(x,z)=\sum_{m=0}^{\infty}\left(a_{m}e^{k_{m}x} + b_{m}e^{-k_{m}x}\right)\chi_{m}(z), \quad x\lt 0. }[/math]

In this expression the term involving [math]\displaystyle{ e^{ikx} }[/math] (recall that [math]\displaystyle{ k_{0} = -ik }[/math]) corresponds to a wave propagating towards large positive [math]\displaystyle{ x }[/math] while the term involving [math]\displaystyle{ e^{-ikx} }[/math] corresponds to a wave propagating towards large negative [math]\displaystyle{ x }[/math]. Terms which decay as [math]\displaystyle{ x \rightarrow -\infty }[/math] or as [math]\displaystyle{ x \rightarrow \infty }[/math] are referred to as evanescent modes.

The expansion of the potential for the region under the dock is

[math]\displaystyle{ \phi(x,z)=\sum_{m=0}^{\infty}\left(a_{m}e^{\kappa_{m}x} + b_{m}e^{-\kappa_{m}x}\right)\psi_{m}(z), \quad x\gt 0. }[/math]

To obtain a unique solution, the diffracted field [math]\displaystyle{ \phi_{\mathrm{D}} }[/math] must satisfy the Sommerfeld radiation condition (see further above) specifying that the waves of this potential propagate away from the structure and the solution be bounded.

Application of the radiation condition to both the general solutions immediately above give diffracted potentials of

[math]\displaystyle{ \phi_{\mathrm{D}}(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\chi_{m}(z), \quad x\lt 0 }[/math]

and

[math]\displaystyle{ \phi_{\mathrm{D}}(x,z)=\sum_{m=0}^{\infty}b_{m} e^{-\kappa_{m}x}\psi_{m}(z), \quad x\gt 0 }[/math]

where [math]\displaystyle{ a_{m} }[/math] and [math]\displaystyle{ b_{m} }[/math] are the coefficients of the potential in the open water and the dock covered region respectively.

Incident potential

To create meaningful solutions of the velocity potential [math]\displaystyle{ \phi }[/math] in the specified domains we add an incident wave term to the expansion for the domain of [math]\displaystyle{ x \lt 0 }[/math] above. The incident potential is a wave of amplitude [math]\displaystyle{ A }[/math] in displacement travelling in the positive [math]\displaystyle{ x }[/math]-direction. We would only see this in the time domain [math]\displaystyle{ \Phi(x,z,t) }[/math] however, in the frequency domain the incident potential can be written as

[math]\displaystyle{ \phi_{\mathrm{I}}(x,z) =e^{-k_{0}x}\chi_{0}\left( z\right). }[/math]

The total velocity (scattered) potential now becomes [math]\displaystyle{ \phi = \phi_{\mathrm{I}} + \phi_{\mathrm{D}} }[/math] for the domain of [math]\displaystyle{ x \lt 0 }[/math].

The first term in the expansion of the diffracted potential for the domain [math]\displaystyle{ x \lt 0 }[/math] is given by

[math]\displaystyle{ a_{0}e^{k_{0}x}\chi_{0}\left( z\right) }[/math]

which represents the reflected wave.

In any scattering problem [math]\displaystyle{ |R|^2 + |T|^2 = 1 }[/math] where [math]\displaystyle{ R }[/math] and [math]\displaystyle{ T }[/math] are the reflection and transmission coefficients respectively. In our case of the semi-infinite dock [math]\displaystyle{ |a_{0}| = |R| = 1 }[/math] and [math]\displaystyle{ |T| = 0 }[/math] as there are no transmitted waves in the region under the dock.

An infinite dimensional system of equations

The potential and its derivative must be continuous across the transition from open water to the dock covered region. Therefore, the potentials and their derivatives at [math]\displaystyle{ x=0 }[/math] have to be equal. We obtain:

[math]\displaystyle{ \begin{align} \chi_{0}\left( z\right) + \sum_{m=0}^{\infty}a_{m} \chi_{m}\left( z\right) &=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z), \quad -h \leq z\leq -d \\ -k_{0}\chi_{0}\left( z\right) +\sum_{m=0}^{\infty} a_{m}k_{m}\chi_{m}\left( z\right) &= \begin{cases} \quad \quad \quad 0,\quad \quad \quad \quad -d\leq z\leq 0 \\ -\sum_{m=1}^{\infty}b_{m}\kappa_{m}\psi_{m}(z),\quad -h\leq z\leq -d \end{cases} \end{align} }[/math]

For the first equation we multiply both sides by [math]\displaystyle{ \psi_{n}(z) \, }[/math] and integrating from [math]\displaystyle{ -h }[/math] to [math]\displaystyle{ -d }[/math] to obtain:

[math]\displaystyle{ B_{n0} + \sum^{\infty}_{m=0}a_{m}B_{nm} = b_{m}C_{m}\delta_{mn}, \quad n\in\mathbb{N}\cup\left\{0\right\} }[/math]

and for the second equation we multiply both sides by [math]\displaystyle{ \chi_{n}(z) \, }[/math] and integrating from [math]\displaystyle{ -h }[/math] to [math]\displaystyle{ 0 }[/math] to obtain:

[math]\displaystyle{ -k_{0}A_{0}\delta_{0n} + k_{m}a_{m}A_{m}\delta_{mn} = -\sum^{\infty}_{m=1}\kappa_{m}b_{m}B_{mn}, \quad n\in\mathbb{N}\cup\left\{0\right\} }[/math]

Remember [math]\displaystyle{ \kappa_{0} = 0 \, }[/math]. Solving the equations above will yield the coefficients of the water velocity potential in the free surface and dock covered regions.

Numerical Solution

To solve the system of equations, we set the upper limit of [math]\displaystyle{ m }[/math] to be [math]\displaystyle{ N }[/math]. This resulting system can be expressed in the block matrix form below,

[math]\displaystyle{ \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 & 0 & \kappa_{1}B_{10} & \cdots & \kappa_{N}B_{N0} \\ 0 & k_{1}A_{1} & \cdots & \vdots & 0 & \kappa_{1}B_{11} & \cdots & \kappa_{N}B_{N1} \\ \vdots & \vdots & \ddots & 0 & \vdots & \vdots & \ddots & \vdots \\ 0 & \cdots & 0 & k_{N}A_{N} & 0 & \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]

We then simply need to solve the linear system of equations.

Matlab Code

A program to calculate the coefficients for the semi-infinite dock problem can be found here semiinfinite_dock.m

Additional code

This program requires