View source for Eigenfunction Matching for a Submerged Semi-Infinite Dock

{{complete pages}}

== Introduction ==

The problems consists of a region to the left 
with a free surface and a region to the right with a free surface 
and a submerged dock/plate through which
not flow is possible. 
We begin with the simply 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 refer to the solution [[Eigenfunction Matching for a Semi-Infinite Dock]].
The solution is a modified version of that given by [[Linton and Evans 1991]].

[[Image:submerged_inf_dock.jpg|thumb|right|300px|Wave scattering by a submerged semi-infinite 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>
\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,\,x>0,
</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. 

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

{{separation of variables for a dock}}
We now adjust for the fact that the dock is submerged.
{{separation of variables for a submerged dock}}

{{free surface submerged plate relations}}

=== Expansion of the potential === 

We need to apply some boundary conditions at plus and minus infinity, 
where are essentially that the solution cannot grow. This means that we
only have the positive (or negative) roots of the dispersion equation.
However, it does not help us with the purely imaginary root. Here we
must use a different condition, essentially identifying one solution
as the incoming wave and the other as the outgoing wave. 

Therefore the potential (without the incident wave, which will
be added later) can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}^{h}x}\phi_{m}^{h}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\mu_{m}x}\chi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the dock covered region respectively.

{{incident potential for two dimensions}}

=== 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=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left(  z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}^{h}\left(  z\right) 
=\sum_{m=0}^{\infty}b_{m}\chi_{m}(z)
</math>
</center>
and
<center>
<math>
 -k_{0}^{h}\phi_{0}\left(  z\right)  +\sum
_{m=0}^{\infty} a_{m}k_{m}^{h}\phi_{m}^{h}\left(  z\right) 
 =-\sum_{m=0}^{\infty}b_{m}\mu_{m}\chi
_{m}(z)
</math>
</center>
for each <math>n</math>.
We solve these equations by multiplying both equations by
<math> \phi_{l}^{h}(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_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-k_{0}^{h}A_{0}\delta_{0l}+a_{l}k_{l}^{h}A_l
 =-\sum_{m=0}^{\infty}b_{m}\mu_{m}B_{ml}
</math>
</center>
We solve these equations in the standard way by truncation.

== 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^{-\hat{k}_0x}\phi_0(z)+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\hat{\mu}_{m}x}\chi_{m}(z), \;\;x>0
</math>
</center>
where <math>\hat{k}_{m} = \sqrt{k_m^2 - k_y^2}</math> and <math>\hat{\mu}_{m} = \sqrt{\mu_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_{n=0}^{\infty}b_{m}B_{ml}
</math>
</center>
and
<center>
<math>
-\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l
 =-\sum_{m=0}^{\infty}b_{m}\hat{\mu}_{m}B_{ml} 
</math>
</center>
and these are solved exactly as before.

== Energy Balance ==

{{energy contour and preliminaries}}

We only get contributions from the vertical ends and we obtain

<center><math>
\lim_{x\to-\infty} \Im\int_{-h}^{0}\phi^*\frac{\partial\phi}{\partial n} \mathrm{d}z  
=  \lim_{x\to-\infty} \Im\int_{-h}^{0} \hat{k}_0 \left(e^{\hat{k}_0 x} + a_0^{*} e^{-\hat{k}_0 x} \right)
\left(e^{-\hat{k}_0 x} - a_0 e^{\hat{k}_0 x} \right)\phi_0(z)^2 \mathrm{d}z  
</math></center>
<center><math>
= \frac{\hat{k}_0}{i} A_0 \left(1 - |a_0^2| \right)
</math></center>
and 
<center><math>
\lim_{x\to\infty} \Im\int_{-h}^{0}\phi^*\frac{\partial\phi}{\partial n} \mathrm{d}z  
=  \lim_{x\to\infty} \Im\int_{-h}^{0} -\hat{\mu}_0 \left(b_0^{*} e^{\hat{\mu}_0 x} \right)
\left(b_0 e^{-\hat{\mu}_0 x}\right)\chi_0(z)^2 \mathrm{d}z  
</math></center>
<center><math>
= - \frac{\hat{\mu}_0}{i} |b_0^2| \int_{-h}^{0} \chi_0(z)^2 \mathrm{d}z  
</math></center>
Therefore the energy balance can be written as
<center><math>
\hat{k}_0 A_0 |a_0^2| + \hat{\mu}_0 |b_0^2| \int_{-h}^{0} \chi_0(z)^2 \mathrm{d}z  = \hat{k}_0 A_0
</math></center>
Note that the second integral above is actually only from <math>-d</math> to 0.

== 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_semiinfinite_dock.m submerged_semiinfinite_dock.m]

=== Additional code ===

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


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