View source for Two Identical Submerged Docks using Symmetry

{{complete pages}}

== Introduction ==

This is the extension of [[Eigenfunction Matching for a Submerged Finite Dock]] using 
[[:Category:Symmetry in Two Dimensions|Symmetry in Two Dimensions]].. 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 [[Two Identical Docks using Symmetry]]

==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>
\phi_{z}=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_2<x<-L_1,\,\,{\rm and}\,\,L_1<x<L_2
</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 three regions, <math>x<0</math>
<math>-d<z<0,\,\,x>0</math>, and <math>-h<z<-d,\,\,x>0</math>. The first two regions use the free-surface eigenfunction
and the third uses the 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^{h}_{0}(x+L_2)}\phi_{0}\left(
z\right) 
</math>
</center>

We use [[:Category:Symmetry in Two Dimensions|Symmetry in Two Dimensions]] and express the symmetric solution
as
<center>
<math>
\phi(x,z)=e^{-k_{0}^h (x+L_2)}\phi_{0}^h\left(
z\right) + \sum_{m=0}^{\infty}a_{m}^{s}e^{k_{m}^h (x+L_2)}\phi_{m}^h(z), \;\;x<-L_2
</math>
</center>
<center>
<math>
\phi(x,z)= \sum_{m=0}^{\infty}b_{m}^{s}
e^{-k_{m}^d (x+L_2)}\phi_{m}^d(z)
+ \sum_{m=0}^{\infty}c_{m}^{s}
e^{k_{m}^d (x+L_1)}\phi_{m}^d(z)
, \;\;-d<z<0,\,\,x<-L_2,\,-L_1<x<L_1, {\rm or} \, x>L_2
</math>
</center>
and
<center>
<math>
\phi(x,z)= d_0^{s}\frac{x+L_1}{L_2-L_1} + \sum_{m=1}^{\infty}d_{m}^{s}
e^{\kappa_{m} (x+L_2)}\psi_{m}(z)
+
e_0^{s}\frac{x+L_2}{L_2-L_1} 
\sum_{m=0}^{\infty}e_{m}^{s}
e^{-\kappa_{m} (x+L_1)}\psi_{m}(z)
, \;\;-h<z<-d,\,\,\,-L_2<x<-L_1, {\rm or} \, L_1<x<L_2,
</math>
</center>
<center>
<math>
\phi(x,z)= \sum_{m=0}^{\infty}f_{m}^{s}\frac{\cosh(k_{m}^h x)}{\cosh(k_m^h L)}\phi_{m}^h(z), \;\;x>L
</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.

The anti-symmetric solution is 
<center>
<math>
\phi(x,z)=e^{-k_{0}^h (x+L_2)}\phi_{0}^h\left(
z\right) + \sum_{m=0}^{\infty}a_{m}^{a}e^{k_{m}^h (x+L_2)}\phi_{m}^h(z), \;\;x<-L_2
</math>
</center>
<center>
<math>
\phi(x,z)= \sum_{m=0}^{\infty}b_{m}^{a}
e^{-k_{m}^d (x+L_2)}\phi_{m}^d(z)
+ \sum_{m=0}^{\infty}c_{m}^{a}
e^{k_{m}^d (x+L_1)}\phi_{m}^d(z)
, \;\;-d<z<0,\,\,x<-L_2,\,-L_1<x<L_1, {\rm or} \, x>L_2
</math>
</center>
and
<center>
<math>
\phi(x,z)= d_0^{a}\frac{x+L_1}{L_2-L_1} + \sum_{m=1}^{\infty}d_{m}^{a}
e^{-\kappa_{m} (x+L_2)}\psi_{m}(z)
+
e_0^{a}\frac{x+L_2}{L_2-L_1} 
\sum_{m=1}^{\infty}e_{m}^{a}
e^{\kappa_{m} (x+L_1)}\psi_{m}(z)
, \;\;-h<z<-d,\,\,\,-L_2<x<-L_1, {\rm or} \, L_1<x<L_2,
</math>
</center>
<center>
<math>
\phi(x,z)= \sum_{m=0}^{\infty}f_{m}^{a}\frac{\sinh(k_{m}^h x)}{\sinh(k_m^h L)}\phi_{m}^h(z), \;\;x>L
</math>
</center>

== Matlab Code ==

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

=== Additional code ===

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


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