View source for Eigenfunction Matching for a Semi-Infinite Dock

{{complete pages}}

== Introduction ==

This is one of the simplest problem in eigenfunction matching. It also is an easy
problem to understand the [[:Category:Wiener-Hopf|Wiener-Hopf]] and 
[[:Category:Residue Calculus|Residue Calculus]].  The problems consists of a region to the left 
with a free surface and a region to the right with a rigid surface 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 give the equations in slightly less detail.
The case of a [[Finite Dock]] is treated very similarly. The problem can also
be generalised to a [[ Eigenfunction Matching for Submerged Semi-Infinite Dock|Submerged Semi-Infinite Dock]]

[[Image:semiinfinite_dock.jpg|thumb|right|300px|Wave scattering by a semi-infinite dock]]

== 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>
\partial_z\phi=0, \,\, z=-h,
</math>
</center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center>
<math>
\partial_z\phi=0, \,\, z=0,\,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 two regions, <math>x<0</math>
and <math>x>0</math>.

{{separation of variables in two dimensions}}

{{separation of variables for a free surface}}

{{separation of variables for a dock}}

{{free surface dock relations}}

=== Expansion of the potential === 

We need to apply some boundary conditions at plus and minus infinity, 
where are essentially the 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 scattered 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}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{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}\left(  z\right) 
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math>
</center>
and
<center>
<math>
 -k_{0}\phi_{0}\left(  z\right)  +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left(  z\right) 
 =-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>m</math>.
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}\,\,\,(3)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
 =-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4)
</math>
</center>
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
we obtain 
<center>
<math>
(k_{0}+k_l)A_{0}\delta_{0l}
 =\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
</math>
</center>
This equation gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

== Numerical Solution ==

To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{ml}&\vdots\\
&&\\
-B_{M0}&\cdots&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&&\\
\vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\
&&\\
(k_0 + \kappa_M) \, B_{M0}&\cdots&(k_M + \kappa_{M}) \, B_{MM}\\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_M \\
\\
b_{0}\\
b_1 \\
\vdots \\
\\
b_M
\end{bmatrix}
=
\begin{bmatrix}
- A_{0} \\
0 \\
\vdots \\
\\
0 \\
\\
2k_{0}A_{0} \\
0 \\
\vdots \\
\\
0
\end{bmatrix}
</math>
</center>
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>. 

{{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{\kappa}_{m}x}\psi_{m}(z), \;\;x>0
</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_{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{\kappa}_{m}B_{ml} 
</math>
</center>
and these are solved exactly as before.

== Matlab Code ==

A program to calculate the coefficients for the semi-infinite dock problems can be found here
{{semiinfinite_dock code}}

=== Additional code ===

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

[[Category:Eigenfunction Matching Method]]