View source for Eigenfunction Matching for a Finite Dock using Symmetry

{{complete pages}}

== Introduction ==

The problems consists of a region to the left and right
with a free surface and a middle region with a rigid surface 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 then consider the case when the waves are incident
at an angle. For the later we give the equations in slightly less detail.
The theory is based on [[Eigenfunction Matching for a Semi-Infinite Dock]] and this should
be consulted for many details.

==Governing Equations==

{{finite dock equations}}

==Solution Method==

We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the three regions, exactly
as for the [[Eigenfunction Matching for a Semi-Infinite Dock]]. 

{{separation of variables for a free surface}}

{{separation of variables for a dock}}

{{free surface dock relations}}

{{incident potential for two dimensions}}

== Expansion of the Potential == 
The potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-k_{0}(x+L)}\phi_{0}\left(
z\right) + \sum_{m=0}^{\infty}a_{m}e^{k_{m}(x+L)}\phi_{m}(z), \;\;x<-L
</math>
</center>
<center>
<math>
\phi(x,z)=b_0 \frac{L-x}{2L}\psi_{0}(z) + \sum_{m=1}^{\infty}b_{m}
e^{-\kappa_{m} (x+L)}\psi_{m}(z)
+c_0 \frac{L+x}{2L}\psi_{0}(z)  + \sum_{m=1}^{\infty}c_{m}
e^{\kappa_{m} (x-L)}\psi_{m}(z)
, \;\;-L<x<L
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}d_{m}
e^{-k_{m}(x-L)}\phi_{m}(z), \;\;L<x
</math>
</center>
where <math>a_{m}</math> and <math>d_{m}</math>
are the coefficients of the potential in the open water regions to the 
left and right and <math>c_m</math> and <math>d_m</math> are the coefficients under the dock
covered region. We have an incident wave from the left. 

== Solution using Symmetry ==

The finite dock problem is symmetric about the line <math>x=0</math> and this allows us to solve the problem
using symmetry. This method is numerically more efficient and requires only slight modification of the 
code for [[Eigenfunction Matching for a Semi-Infinite Dock]], the developed theory here is very close
to the semi-infinite solution. 
We decompose the solution into a symmetric and an anti-symmetric part as is described in
[[:Category:Symmetry in Two Dimensions|Symmetry in Two Dimensions]]

=== Symmetric solution ===
The symmetric potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-k_{0}(x+L)}\phi_{0}\left(
z\right) + \sum_{m=0}^{\infty}a_{m}^{s}e^{k_{m}(x+L)}\phi_{m}(z)
, \;\;x<-L
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}^{s}
\frac{\cosh\kappa_{m} x}{\cosh \kappa_m L}\phi_{m}(z), \;\;-L<x<0
</math>
</center>
where <math>a_{m}^{s}</math> and <math>b_{m}^{s}</math>
are the coefficients of the potential in the open water regions and the dock
covered region respectively.

We now match at <math>x=-L</math> and multiply 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}^{s}A_{l}
=\sum_{n=0}^{\infty}b_{m}^{s}B_{ml}
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}^{s}k_{l}A_l
 =-\sum_{m=0}^{\infty}b_{m}^{s}\kappa_{m}\tanh(\kappa_{m}L) B_{ml}
</math>
</center>
(for full details of this derivation see [[Eigenfunction Matching for a Semi-Infinite Dock]])



=== Anti-symmetric solution ===

The anti-symmetric potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-k_{0}(x+L)}\phi_{0}\left(
z\right) + \sum_{m=0}^{\infty}a_{m}^{a}e^{k_{m}(x+L)}\phi_{m}(z)
, \;\;x<-L
</math>
</center>
and
<center>
<math>
\phi(x,z)=-b_0^{a}\frac{x}{L}+\sum_{m=1}^{\infty}b_{m}^{a}
\frac{\sinh\kappa_{m} x}{-\sinh \kappa_m L}\phi_{m}(z), \;\;-L<x<0
</math>
</center>
where <math>a_{m}^{a}</math> and <math>b_{m}^{a}</math>
are the coefficients of the potential in the open water regions and the dock
covered region respectively. Note that the minus sign in the expression for the dock-covered region has been added so that each component is equal to one at <math>x=-L</math>.

We now match at <math>x=-L</math> and multiply 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}A_{l}
=\sum_{n=0}^{\infty}b_{m}^{a}B_{ml}
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}^{a}k_{l}A_l
 = - \frac{b_{0}^{a}}{L}B_{0l} -\sum_{m=1}^{\infty}b_{m}^{a}\kappa_{m}\coth(\kappa_{m}L) B_{ml}
</math>
</center>

(for full details of this derivation see [[Eigenfunction Matching for a Semi-Infinite Dock]])

=== Solution to the original problem ===

We can now reconstruct the potential for the finite dock from the two previous symmetric and anti-symmetric solution as explained in 
[[:Category:Symmetry in Two Dimensions|Symmetry in Two Dimensions]].
The amplitude in the left open-water region is simply obtained by the superposition principle
<center>
<math>
a_{m} = \frac{1}{2}\left(a_{m}^{s}+a_{m}^{a}\right)
</math>
</center>
<center>
<math>
d_{m} = \frac{1}{2}\left(a_{m}^{s}-a_{m}^{a}\right)
</math>
</center>
Note the formulae for <math>b_m</math> and <math>c_m</math>
are more complicated but can be derived with some work.

== 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^{-k_{0}(x+L)}\phi_{0}\left(
z\right) + \sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}(x+L)}\phi_{m}(z), \;\;x<-L
</math>
</center>
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\kappa}_{m} (x+L)}\psi_{m}(z)
++ \sum_{m=1}^{\infty}c_{m}
e^{\hat{\kappa}_{m} (x-L)}\psi_{m}(z)
, \;\;-L<x<L
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}d_{m}
e^{-\hat{k}_{m}(x-L)}\phi_{m}(z), \;\;L<x
</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.

== Matlab Code ==

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

=== Additional Code ===
This program requires
* {{free surface dispersion equation code}}

[[Category:Eigenfunction Matching Method]]
[[Category:Pages with Matlab Code]]
[[Category:Symmetry in Two Dimensions]]
[[Category:Complete Pages]]