View source for Two Identical Docks using Symmetry

= Introduction =

The problems consists three regions 
with a free surface and and two regions of identical length with a rigid surface through which
not flow is possible. 
The solution method is an extension of [[Eigenfunction Matching for a Finite Dock]]
using [[:Category:Symmetry in Two Dimensions|Symmetry in Two Dimensions]].
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.

=Governing Equations=

We begin with the [[Frequency Domain Problem]] for a dock which occupies
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,\,x<-L_2,\,-L_1<x<L_1, {\rm or} \, x>L_2
</math></center>
<center>
<math>
\partial_z\phi=0, \,\, z=0,\,-L_2<x<-L_1, {\rm or} \, L_1<x<L_2,
</math>
</center>
where we require <math>L_1<L_2</math> and we define <math>L_2 - L_1 = 2L</math>
(so that the dock also has length <math>2L</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=

The solution method uses [[Symmetry in Two Dimensions]] and we write the potential as
a symmetric and an anti-symmetric part and consider only the region <math>x<0</math>. We apply
either Neuman (symmetric) or Dirichlet (anti-symmetric) boundary conditions at <math>x=0</math>.
We [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the three regions, similar to
as for the [[Eigenfunction Matching for a Finite Dock]]. We being with the symmetric potential which can
be expanded as
<center>
<math>
\phi^s(x,z)=e^{-k_{0}(x+L_2)}\phi_{0}\left(
z\right) + \sum_{m=0}^{\infty}a^s_{m}e^{k_{m}(x+L_2)}\phi_{m}(z), \;\;x<-L_2
</math>
</center>
<center>
<math>
\phi(x,z)=b^s_0 \frac{x+L_1}{-2L}\psi_{0}(z) + \sum_{m=1}^{\infty}b^s_{m}
e^{-\kappa_{m} (x+L_2)}\psi_{m}(z)
+c^s_0 \frac{L_2+x}{2L}\psi_{0}(z)  + \sum_{m=1}^{\infty}c^s_{m}
e^{\kappa_{m} (x+L_1)}\psi_{m}(z)
, \;\;-L_2<x<L_1
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}d^s_{m} \frac{\cosh(k_{m}x)}{\cosh(k_m L_1)} \phi_{m}(z), \;\;-L_1<x<0
</math>
</center>
where <math>a^s_{m}</math> and <math>d^s_{m}</math>
are the coefficients of the potential in the open water regions to the 
left and right and <math>b^s_m</math> and <math>c^s_m</math> are the coefficients under the dock. 
<math>k_n</math> are the roots of the 
[[Dispersion Relation for a Free Surface]]. We denote the
positive imaginary solutions by <math>k_{0}</math> and
the positive real solutions by <math>k_{m}</math>, <math>m\geq1</math> (ordered with increasing
imaginary part) and
<math>\kappa_{m}=m\pi/h</math>. We define
<center>
<math>
\phi_{m}\left(  z\right)  = \frac{\cos k_{m}(z+h)}{\cos k_{m}h},\quad m\geq0
</math>
</center>
as the vertical eigenfunction of the potential in the open
water region and
<center>
<math>
\psi_{m}\left(  z\right)  = \cos\kappa_{m}(z+h),\quad
m\geq 0
</math>
</center>
as the vertical eigenfunction of the potential in the dock
covered region. For later reference, we note that:
<center>
<math>
\int\nolimits_{-h}^{0}\phi_{m}(z)\phi_{n}(z) d z=A_{m}\delta_{mn}
</math>
</center>
where
<center>
<math>
A_{m}=\frac{1}{2}\left(  \frac{\cos k_{m}h\sin k_{m}h+k_{m}h}{k_{m}\cos
^{2}k_{m}h}\right)
</math>
</center>
and
<center>
<math>
\int\nolimits_{-h}^{0}\phi_{n}(z)\psi_{m}(z) d z=B_{mn}
</math>
</center>
where
<center><math>
B_{mn}=\frac{k_{n}\sin k_{n}h\cos\kappa_{m}h-\kappa_{m}\cos k_{n}h\sin
\kappa_{m}h}{\left(  \cos k_{n}h\cos\kappa_{m}h\right)  \left(  k_{n}
^{2}-\kappa_{m}^{2}\right)  }
</math></center>
and
<center>
<math>
\int\nolimits_{-h}^{0}\psi_{m}(z)\psi_{n}(z) d z=C_{m}\delta_{mn}
</math>
</center>
where
<center>
<math>
C_{m}=\frac{1}{2}h,\quad,m\neq 0 \quad \mathrm{and} \quad C_0 = h
</math></center>

==An infinite dimensional system of equations==

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=- L_2</math> and
<math>x=-L_1</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left(  z\right) + \sum_{m=0}^{\infty}
a^s_{m} \phi_{m}\left(  z\right) 
=\sum_{m=0}^{\infty}b^s_{m}\psi_{m}(z) + \sum_{m=1}^{\infty}c^s_{m}\psi_{m}(z)e^{-2L\kappa_m}
</math>
</center>
<center>
<math>
 -k_{0}\phi_{0}\left(  z\right)  +\sum
_{m=0}^{\infty} a^s_{m}k_{m}\phi_{m}\left(  z\right) 
 =-\frac{b^s_0}{2L}\psi_0(z) -\sum_{m=1}^{\infty}b^s_{m}\kappa_{m}\psi
_{m}(z) + \frac{c^s_0}{2L}\psi_0(z)+\sum_{m=1}^{\infty}c^s_{m}\kappa_{m}\psi
_{m}(z)e^{-2L\kappa_m}
</math>
</center>
<center>
<math>
\sum_{m=1}^{\infty}b^s_{m}\psi_{m}(z)e^{-2L\kappa_m} + \sum_{m=0}^{\infty}c^s_{m}\psi_{m}(z)
=\sum_{m=0}^{\infty}d^s_{m} \phi_{m}\left(  z\right) 
</math>
</center>
<center>
<math>
-\frac{b^s_0}{2L}\psi_0(z) -\sum_{m=1}^{\infty}b^s_{m}\kappa_{m}\psi
_{m}(z)e^{-2L\kappa_m} + \frac{c^s_0}{2L}\psi_0(z)+\sum_{m=1}^{\infty}c^s_{m}\kappa_{m}\psi
_{m}(z)
= \sum_{m=0}^{\infty}d^s_{m} \tanh(k_m L_1) k_m\phi_{m}\left(  z\right) 
</math>
</center>

We solve these equations by multiplying 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^s_{l}A_{l}
=\sum_{m=0}^{\infty}b^s_{m}B_{ml}
+ \sum_{m=1}^{\infty}c^s_{m}B_{ml}e^{-2L\kappa_m}
</math>
</center>
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a^s_{l}k_{l}A_l
 = - b^s_0 \frac{B_{0l}}{2L} - \sum_{m=1}^{\infty}b^s_{m}\kappa_{m}B_{ml} 
+ c^s_0 \frac{B_{0l}}{2L} + \sum_{m=1}^{\infty}c^s_{m}\kappa_{m}B_{ml} e^{-2L\kappa_m}
</math>
</center>
<center>
<math>
\sum_{m=1}^{\infty}b^s_{m}B_{ml}e^{-2L\kappa_m}
+ \sum_{m=0}^{\infty}c^s_{m}B_{ml}
=d^s_l A_l
</math>
</center>
<center>
<math>
- b^s_0 \frac{B_{0l}}{2L} - \sum_{m=1}^{\infty}b^s_{m}\kappa_{m}B_{ml} e^{-2L\kappa_m}
+ c^s_0 \frac{B_{0l}}{2L} + \sum_{m=1}^{\infty}c^s_{m}\kappa_{m}B_{ml} 
= d^s_l \tanh(k_l L_1) k_l A_l
</math>
</center>

=Numerical Solution=

To solve the system of equations we set the upper limit of <math>l</math> to
be <math>M</math>. We then simply need to solve the linear system of equations.

= Anti-Symmetric Solution = 

The solution for the anti-symmetric potential proceeds in an almost identical manner. 

<center>
<math>
\phi^a(x,z)=e^{-k_{0}(x+L_2)}\phi_{0}\left(
z\right) + \sum_{m=0}^{\infty}a^a_{m}e^{k_{m}(x+L_2)}\phi_{m}(z), \;\;x<-L_2
</math>
</center>
<center>
<math>
\phi(x,z)=b^a_0 \frac{x+L_1}{-2L}\psi_{0}(z) + \sum_{m=1}^{\infty}b^a_{m}
e^{-\kappa_{m} (x+L_2)}\psi_{m}(z)
+c^a_0 \frac{L_2+x}{2L}\psi_{0}(z)  + \sum_{m=1}^{\infty}c^a_{m}
e^{\kappa_{m} (x+L_1)}\psi_{m}(z)
, \;\;-L_2<x<L_1
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}d^a_{m} \frac{\sinh(k_{m}x)}{-\sinh(k_m L_1)} \phi_{m}(z), \;\;-L_1<x<0
</math>
</center>
where <math>a^a_{m}</math> and <math>d^a_{m}</math>
are the coefficients of the potential in the open water regions to the 
left and right and <math>b^a_m</math> and <math>c^a_m</math> are the coefficients under the dock.

We solve these equations by multiplying 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^a_{l}A_{l}
=\sum_{m=0}^{\infty}b^a_{m}B_{ml}
+ \sum_{m=1}^{\infty}c^a_{m}B_{ml}e^{-2L\kappa_m}
</math>
</center>
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a^a_{l}k_{l}A_l
 = - b^a_0 \frac{B_{0l}}{2L} - \sum_{m=1}^{\infty}b^a_{m}\kappa_{m}B_{ml} 
+ c^a_0 \frac{B_{0l}}{2L} + \sum_{m=1}^{\infty}c^a_{m}\kappa_{m}B_{ml} e^{-2L\kappa_m}
</math>
</center>
<center>
<math>
\sum_{m=1}^{\infty}b^a_{m}B_{ml}e^{-2L\kappa_m}
+ \sum_{m=0}^{\infty}c^a_{m}B_{ml}
=d^a_l A_l
</math>
</center>
<center>
<math>
- b^a_0 \frac{B_{0l}}{2L} - \sum_{m=1}^{\infty}b^a_{m}\kappa_{m}B_{ml} e^{-2L\kappa_m}
+ c^a_0 \frac{B_{0l}}{2L} + \sum_{m=1}^{\infty}c^a_{m}\kappa_{m}B_{ml} 
= d^a_l  \frac{k_l A_l}{\tanh(k_l L_1)}
</math>
</center>

= Solution with Waves Incident at an Angle =

We can consider the problem when the waves are incident at an angle <math>\theta</math>. In this 
case we have the wavenumber in the <math>y</math> direction is <math>k_y = \sin\theta k_0</math>
where <math>k_0</math> is as defined previously (note that <math>k_y</math> is imaginary). In some 
ways the solution is now simpler because we do not need to write the zero term separately
under the dock. 

This means that the potential is now of the form <math>\phi(x,y,z)=e^{k_y y}\phi(x,z)</math>.
Therefore the symmetric potential can
be expanded as
<center>
<math>
\phi^{s}_{0}\left(  z\right) + \sum_{m=0}^{\infty}
a^s_{m} \phi_{m}\left(  z\right) 
=\sum_{m=0}^{\infty}b^s_{m}\psi_{m}(z) + \sum_{m=0}^{\infty}c^s_{m}\psi_{m}(z)e^{-2L\hat{\kappa}_m}
</math>
</center>
<center>
<math>
 -\hat{k}_{0}\phi_{0}\left(  z\right)  +\sum
_{m=0}^{\infty} a^s_{m}\hat{k}_{m}\phi_{m}\left(  z\right) 
 = -\sum_{m=0}^{\infty}b^s_{m}\hat{\kappa}_{m}\psi
_{m}(z) +\sum_{m=0}^{\infty}c^s_{m}\hat{\kappa}_{m}\psi
_{m}(z)e^{-2L\hat{\kappa}_m}
</math>
</center>
<center>
<math>
\sum_{m=1}^{\infty}b^s_{m}\psi_{m}(z)e^{-2L\hat{\kappa}_m} + \sum_{m=0}^{\infty}c^s_{m}\psi_{m}(z)
=\sum_{m=0}^{\infty}d^s_{m} \phi_{m}\left(  z\right) 
</math>
</center>
<center>
<math>
-\sum_{m=0}^{\infty}b^s_{m}\hat{\kappa}_{m}\psi
_{m}(z)e^{-2L\hat{\kappa}_m} + \sum_{m=0}^{\infty}c^s_{m}\hat{\kappa}_{m}\psi
_{m}(z)
= \sum_{m=0}^{\infty}d^s_{m} \tanh(\hat{k}_m L_1) \hat{k}_m\phi_{m}\left(  z\right) 
</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>
and we always take the positive real root or the root with positive imaginary part. 

We solve these equations by multiplying 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^s_{l}A_{l}
=\sum_{m=0}^{\infty}b^s_{m}B_{ml}
+ \sum_{m=0}^{\infty}c^s_{m}B_{ml}e^{-2L\hat{\kappa}_m}
</math>
</center>
<center>
<math>
-\hat{k}_{0}A_{0}\delta_{0l}+a^s_{l}\hat{k}_{l}A_l
 = - \sum_{m=0}^{\infty}b^s_{m}\hat{\kappa}_{m}B_{ml} 
+  \sum_{m=0}^{\infty}c^s_{m}\hat{\kappa}_{m}B_{ml} e^{-2L\hat{\kappa}}
</math>
</center>
<center>
<math>
\sum_{m=0}^{\infty}b^s_{m}B_{ml}e^{-2L\hat{\kappa}_m}
+ \sum_{m=0}^{\infty}c^s_{m}B_{ml}
=d^s_l A_l
</math>
</center>
<center>
<math>
- \sum_{m=0}^{\infty}b^s_{m}\hat{\kappa}_{m}B_{ml} e^{-2L\hat{\kappa}_m}
+  \sum_{m=0}^{\infty}c^s_{m}\hat{\kappa}_{m}B_{ml} 
= d^s_l \tanh(\hat{k}_l L_1) \hat{k}_l A_l
</math>
</center>
and these are solved exactly as before. The solution for the anti-symmetric 
potential is found in a similar fashion.

= 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/two_finite_docks_symmetry.m two_finite_docks_symmetry.m]

== Additional code ==

This program requires
[http://www.math.auckland.ac.nz/~meylan/code/dispersion/dispersion_free_surface.m dispersion_free_surface.m]
to run

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