View source for Eigenfunction Matching for a Submerged Finite Dock

{{complete pages}}


= Introduction =

This is the finite length version of the [[Eigenfunction Matching for a Submerged Semi-Infinite Dock]]. The full
theory is not presented here, and details of the matching method can be found in 
[[Eigenfunction Matching for a Submerged Semi-Infinite Dock]] and [[Eigenfunction Matching for a Finite Dock]]

=Governing Equations=

We begin with the [[Frequency Domain Problem]] for the submerged dock in
the region <math>x>0</math>  (we assume <math>e^{i\omega t}</math> time dependence).
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,
</math></center>
<center>
<math>
\partial_z\phi=0, \,\, z=-d,\,-L<x<L,
</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 four regions, {<math>x<-L \,</math>}, {<math>x>L \,</math>}, {<math>-d<z<0,\,\,-L<x<L</math>}, and {<math>-h<z<-d,\,\,-L<x<L</math>}. The first three regions use the free-surface eigenfunction
and the last uses dock eigenfunctions. Details can be found in [[Eigenfunction Matching for a Semi-Infinite Dock]].

The incident potential is a wave of amplitude <math>A</math>
in displacement travelling in the positive <math>x</math>-direction. 
The incident potential can therefore be written as
<center>
<math>
\phi^{\mathrm{I}}  =e^{-k_{0}^{h}(x+L)}\phi_{0}\left(
z\right) 
</math>
</center>

The potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-k_{0}^h (x+L)}\phi_{0}^h\left(
z\right) + \sum_{m=0}^{\infty}a_{m}e^{k_{m}^h (x+L)}\phi_{m}^h(z), \;\;x<-L
</math>
</center>
<center>
<math>
\phi(x,z)= \sum_{m=0}^{\infty}b_{m}
e^{-k_{m}^d (x+L)}\phi_{m}^d(z)
+ \sum_{m=0}^{\infty}c_{m}
e^{k_{m}^d (x-L)}\phi_{m}^d(z)
, \;\;-d<z<0,\,\,-L<x<L
</math>
</center>
and
<center>
<math>
\phi(x,z)= d_0 \frac{L-x}{2 L} + \sum_{m=1}^{\infty}d_{m}
e^{-\kappa_{m} (x+L)}\psi_{m}(z)
+ e_0 \frac{x+L}{2 L} + 
\sum_{m=1}^{\infty}e_{m}
e^{\kappa_{m} (x-L)}\psi_{m}(z)
, \;\;-h<z<-d,\,\,-L<x<L
</math>
</center>
<center>
<math>
\phi(x,z)= \sum_{m=0}^{\infty}f_{m}e^{-k_{m}^h (x-L)}\phi_{m}^h(z), \;\;L<x
</math>
</center>

The definition of all terms can be found in [[Eigenfunction Matching for Submerged Semi-Infinite Dock]],
as can the solution method and the method to extend the solution to waves incident at an angle.

= Matlab Code =

A program to calculate the coefficients for the submerged semi-infinite dock problems can be found here
[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/submerged_finite_dock.m submerged_finite_dock.m]

== Additional code ==

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


[[Category:Eigenfunction Matching Method]]
[[Category:Pages with Matlab Code]]
[[Category:Complete Pages]]