View source for Eigenfunction Matching for a Finite Dock

= 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. The solution here can be straightforwardly extended
using [[Symmetry in Two Dimensions]] to two docks of the same length and this can be
found [[Two Identical Docks using Symmetry]]. We also show how the solution 
can be found using [[Symmetry in Two Dimensions]] for the finite dock. 

[[Image:finite_dock.jpg|thumb|right|300px|Wave scattering by a finite dock]]

=Governing Equations=

We consider here the [[Frequency Domain Problem]] for a finite dock which occupies
the region <math>-L<x<L</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, {\rm or} \, x>L
</math></center>
<center>
<math>
\partial_z\phi=0, \,\, z=0,\,-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 three regions, exactly
as for the [[Eigenfunction Matching for a Semi-Infinite Dock]]. 
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. 
<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=\pm L</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left(  z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left(  z\right) 
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z) + \sum_{m=1}^{\infty}c_{m}\psi_{m}(z)e^{-2L\kappa_m}
</math>
</center>
<center>
<math>
 -k_{0}\phi_{0}\left(  z\right)  +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left(  z\right) 
 =-\frac{b_0}{2L}\psi_0(z) -\sum_{m=1}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z) + \frac{c_0}{2L}\psi_0(z)+\sum_{m=1}^{\infty}c_{m}\kappa_{m}\psi
_{m}(z)e^{-2L\kappa_m}
</math>
</center>
<center>
<math>
\sum_{m=1}^{\infty}b_{m}\psi_{m}(z)e^{-2L\kappa_m} + \sum_{m=0}^{\infty}c_{m}\psi_{m}(z)
=\sum_{m=0}^{\infty}d_{m} \phi_{m}\left(  z\right) 
</math>
</center>
<center>
<math>
-\frac{b_0}{2L}\psi_0(z) -\sum_{m=1}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)e^{-2L\kappa_m} + \frac{c_0}{2L}\psi_0(z)+\sum_{m=1}^{\infty}c_{m}\kappa_{m}\psi
_{m}(z)
= -\sum_{m=0}^{\infty}d_{m} 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_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}
+ \sum_{m=1}^{\infty}c_{m}B_{ml}e^{-2L\kappa_m}
</math>
</center>
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
 = - b_0 \frac{B_{0l}}{2L} - \sum_{m=1}^{\infty}b_{m}\kappa_{m}B_{ml} 
+ c_0 \frac{B_{0l}}{2L} + \sum_{m=1}^{\infty}c_{m}\kappa_{m}B_{ml} e^{-2L\kappa_m}
</math>
</center>
<center>
<math>
\sum_{m=1}^{\infty}b_{m}B_{ml}e^{-2L\kappa_m}
+ \sum_{m=0}^{\infty}c_{m}B_{ml}
=d_l A_l
</math>
</center>
<center>
<math>
- b_0 \frac{B_{0l}}{2L} - \sum_{m=1}^{\infty}b_{m}\kappa_{m}B_{ml} e^{-2L\kappa_m}
+ c_0 \frac{B_{0l}}{2L} + \sum_{m=1}^{\infty}c_{m}\kappa_{m}B_{ml} 
= -d_l 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.

= 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>
so that when we separate variables we obtain 

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 and we obtain
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=0}^{\infty}b_{m}B_{ml}
+ \sum_{m=0}^{\infty}c_{m}B_{ml}e^{-2L\hat{\kappa}_m}
</math>
</center>
<center>
<math>
-\hat{k}_{0}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
 =  - \sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml} 
+ \sum_{m=0}^{\infty}c_{m}\hat{\kappa}_{m}B_{ml} e^{-2L\hat{\kappa}_m}
</math>
</center>
<center>
<math>
\sum_{m=0}^{\infty}b_{m}B_{ml}e^{-2L\hat{\kappa}_m}
+ \sum_{m=0}^{\infty}c_{m}B_{ml}
=d_l A_l
</math>
</center>
<center>
<math>
-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml} e^{-2L\hat{\kappa}_m}
+ \sum_{m=0}^{\infty}c_{m}\hat{\kappa}_{m}B_{ml} 
= -d_l \hat{k}_l A_l
</math>
</center>
and these are solved exactly as before.

= 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.m finite_dock.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]]


= 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
[[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{\sin\kappa_{m} x}{-\sin \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} -\sum_{m=1}^{\infty}b_{m}^{a}\kappa_{m}\cot(\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 [[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>
and in the dock-covered region we now consider a potential written as
<center>
<math>
\phi(x,z)=b_0 \psi_{0}(z) + \sum_{m=1}^{\infty}b_{m}
\frac{\cos\kappa_{m} x}{\cos \kappa_m L}\psi_{m}(z)
-c_0 \frac{x}{L} \psi_{0}(z)  + \sum_{m=1}^{\infty}c_{m}
\frac{\sin\kappa_{m} x}{-\sin \kappa_m L} \psi_{m}(z)
, \;\;-L<x<L
</math>
</center>
and we obtain
<center>
<math>
b_{m}=\frac{1}{2}\left(b_{m}^{s}+b_{m}^{a}\right)
</math>.
</center>
<center>
<math>
c_{m}=\frac{1}{2}\left(b_{m}^{s}-b_{m}^{a}\right)
</math>.
</center>
Finally, in the right open-water region
<center>
<math>
d_{m} = \frac{1}{2}\left(a_{m}^{s}-a_{m}^{a}\right)
</math>
</center>