View source for Eigenfunction Matching for a Semi-Infinite Change in Depth

{{complete pages}}

== Introduction ==

The problem consists of a region to the left with a free water surface with depth <math>d</math>
and to the right a region of depth <math>h</math>. The problem with a finite change in depth is treated in
[[Eigenfunction Matching for a Finite Change in Depth]]

== 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^{-\mathrm{i}\omega t}</math> time dependence).
The depth of is constant <math>h</math> for <math>x<0</math> and
constant <math>d</math> for <math>x>0</math>.
The <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</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, \,\,  -d< z <0, \,\, x>0
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,
</math></center>
<center><math>
\partial_x\phi=0, \,\, -d< z <-h,\,x=0,
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-h,\, x<0
</math></center>
<center><math>
\partial_z\phi=0, \,\, z=-d,\, 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}}

=== 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}^{h} x}\phi_{m}^{h}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-k_{m}^{d}x}\phi_{m}^{d}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the left and right respectively
and <math>k_{m}^{h}</math> denotes the solution for depth <math>h</math> etc.

{{incident potential for two dimensions}}

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


The potential and its derivative must be continuous across <math>x=0</math>. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal or equal to zero 
as appropriate.
We obtain:
<center><math>
\begin{align}
\phi_{0}^{h}\left(  z\right) + \sum_{m=0}^{\infty}a_{m} \phi_{m}^{h}\left(  z\right) 
&=\sum_{m=0}^{\infty}b_{m}\phi_{m}^{d}(z), \quad -h< z <0 \\
-\sum_{m=0}^{\infty}b_{m}k_{m}^{d}\phi_{m}^{d}(z)=  &
\begin{cases}
-k_{0}^{h}\phi_{0}^{h}\left(  z\right) +\sum_{m=0}^{\infty} a_{m}k_{m}^{h}\phi_{m}^{h}\left(  z\right),\quad -h< z <0 \\
0,\quad -d< z <-h
\end{cases}
\end{align}
</math></center>

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

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

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

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite change in depth can be found here
{{semiinfinite_change_in_depth code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]