View source for Eigenfunction Matching for a Semi-Infinite Rectangle

{{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 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>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\leq z\leq 0, \,\, x\leq 0,
</math></center>
<center><math>
\Delta\phi=0, \,\,  -h\leq z\leq -d, \,\, x\geq 0.
</math></center>

<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>

<center><math>
\partial_x\phi=0, \,\, -d\leq z\leq 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>

Where <math>\alpha = \omega^2/g</math>. 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\leq 0</math>
and <math>x\geq 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 partially 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>
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 ===

<center><math>
B_{mn} = \int\nolimits_{-h}^{-d}\psi_{m}(z)\chi_{n}(z) \mathrm{d} z , \quad m\geq1, \quad n\geq0,
</math></center>
where
<center><math>
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></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>

=== 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
<center>
<math>
X^{\prime\prime} - k^2 X = 0,
</math>
</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>
We need to apply some boundary conditions at plus and minus infinity, 
where essentially 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 roots. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave. 

Therefore the diffracted potentials can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\chi_{m}(z), \quad x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \quad 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 dock covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain:
<center><math>
\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 -d\leq z\leq 0 \\
-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi_{m}(z),\quad -h\leq z\leq -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>\chi_{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.

== 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]]