Difference between revisions of "Eigenfunction Matching for a Semi-Infinite Dock"

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= Introduction =
+
{{complete pages}}
 +
 
 +
== Introduction ==
  
 
This is one of the simplest problem in eigenfunction matching. It also is an easy
 
This is one of the simplest problem in eigenfunction matching. It also is an easy
Line 10: Line 12:
 
at an angle. For the later we give the equations in slightly less detail.
 
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
 
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|semi-infinite submerged dock]]
+
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=
+
== Governing Equations ==
  
 
We begin with the [[Frequency Domain Problem]] for a dock which occupies
 
We begin with the [[Frequency Domain Problem]] for a dock which occupies
Line 22: Line 26:
 
<center>
 
<center>
 
<math>
 
<math>
\Delta\phi=0, \,\, -h<z<0,
+
\Delta\phi=0, \,\, -h < z < 0,
 
</math>
 
</math>
 
</center>
 
</center>
 
<center>
 
<center>
 
<math>
 
<math>
\phi_{z}=0, \,\, z=-h,
+
\partial_z\phi=0, \,\, z=-h,
 
</math>
 
</math>
 
</center>
 
</center>
Line 35: Line 39:
 
<center>
 
<center>
 
<math>
 
<math>
\partial_z\phi=0, \,\, z=0,\,x<0,
+
\partial_z\phi=0, \,\, z=0,\,x>0,
 
</math>
 
</math>
 
</center>
 
</center>
Line 44: Line 48:
 
and a wave propagating away.
 
and a wave propagating away.
  
=Solution Method=
+
== Solution Method ==
  
 
We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>x<0</math>
 
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>.
 
and <math>x>0</math>.
 
== Separation of variables==
 
  
We now separate variables and write the potential as
+
{{separation of variables in two dimensions}}
<center>
+
 
<math>
+
{{separation of variables for a free surface}}
\phi(x,z)=\zeta(z)\rho(x)
+
 
</math>
+
{{separation of variables for a dock}}
</center>
+
 
Applying Laplace's equation we obtain
+
{{free surface dock relations}}
<center>
+
 
<math>
+
=== Expansion of the potential ===  
\zeta_{zz}+\mu^{2}\zeta=0.
+
 
</math>
+
We need to apply some boundary conditions at plus and minus infinity,  
</center>
+
where are essentially the the solution cannot grow. This means that we
We then use the boundary condition at <math>z=-h</math>, which is
+
only have the positive (or negative) roots of the dispersion equation.
the same for all <math>x</math> to write
+
However, it does not help us with the purely imaginary root. Here we
<center>
+
must use a different condition, essentially identifying one solution
<math>
+
as the incoming wave and the other as the outgoing wave.  
\zeta=\cos\mu(z+h)
 
</math>
 
</center>
 
where the separation constant <math>\mu^{2}</math> must
 
satisfy different equations depending on whether <math>x>0</math>
 
or <math>x<0</math>. For <math>x<0</math> the boundary condition
 
<center><math>
 
k\tan\left(  kh\right)  =-\alpha,\quad x<0\,\,\,(1)
 
</math></center>
 
which is the [[Dispersion Relation for a Free Surface]]
 
and for <math>x>0</math>
 
<center>
 
<math>
 
\kappa\tan(\kappa h)=0,\quad
 
x>0 \,\,\,(2)
 
</math>
 
</center>
 
Note that we have set <math>\mu=k</math> under the free
 
surface and <math>\mu=\kappa</math> under the plate. We denote the
 
positive imaginary solution of (1) by <math>k_{0}</math> and
 
the positive real solutions by <math>k_{m}</math>, <math>m\geq1</math>. The solutions of
 
(2) are
 
<math>\kappa_{m}=m\pi/h</math>, <math>m\geq 0</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>
 
  
Therefore the potential can
+
Therefore the scattered potential (without the incident wave, which will
 +
be added later) can
 
be expanded as
 
be expanded as
 
<center>
 
<center>
Line 159: Line 89:
 
the dock covered region respectively.
 
the dock covered region respectively.
  
==Incident potential==
+
{{incident potential for two dimensions}}
 
 
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_{0}x}\phi_{0}\left(
 
z\right)
 
</math>
 
</center>
 
  
==An infinite dimensional system of equations==
+
=== An infinite dimensional system of equations ===
  
  
Line 194: Line 114:
 
</math>
 
</math>
 
</center>
 
</center>
for each <math>n</math>.
+
for each <math>m</math>.
 
We solve these equations by multiplying both equations by
 
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:
+
<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
 
<center>
 
<center>
 
<math>
 
<math>
 
A_{0}\delta_{0l}+a_{l}A_{l}
 
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
+
=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
 
</math>
 
</math>
 
</center>
 
</center>
Line 210: Line 130:
 
</math>
 
</math>
 
</center>
 
</center>
If we mutiple equation (3) by <math>k_l</math> and subtract equation (4)
+
If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
 
we obtain  
 
we obtain  
 
<center>
 
<center>
 
<math>
 
<math>
2k_{0}A_{0}\delta_{0l}
+
(k_{0}+k_l)A_{0}\delta_{0l}
  =\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml}  
+
  =\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5)
 
</math>
 
</math>
 
</center>
 
</center>
Line 221: Line 141:
 
coefficients of the water velocity potential in the dock covered region.
 
coefficients of the water velocity potential in the dock covered region.
  
=Numerical Solution=
+
== Numerical Solution ==
  
To solve the system of equations (10) we set the upper limit of <math>l</math> to
+
To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
be <math>M</math>. In terms of matrix, we obtain
+
be <math>M</math>. This resulting system can be expressed in the block matrix form below,
 
<center>
 
<center>
 
<math>
 
<math>
 
\begin{bmatrix}
 
\begin{bmatrix}
 
\begin{bmatrix}
 
\begin{bmatrix}
A_0&0&\cdots&&0\\
+
A_0&0 \quad \cdots&0\\
0&\ddots&\ddots&&\vdots\\
+
0&&\\
\vdots&\ddots&A_l\delta_{0l}&&\\
+
\vdots&A_l&\vdots\\
&&&\ddots&0\\
+
&&0\\
0&\cdots&&0&A_M
+
0&\cdots \quad 0 &A_M
 
\end{bmatrix}
 
\end{bmatrix}
 
&
 
&
 
\begin{bmatrix}
 
\begin{bmatrix}
-B_{00}&-B_{01}&\cdots&&-B_{0M}\\
+
-B_{00}&\cdots&-B_{0M}\\
-B_{10}&\ddots&&&\vdots\\
+
&&\\
\vdots&&-B_{nl}&&\\
+
\vdots&-B_{ml}&\vdots\\
&&&\ddots&\\
+
&&\\
-B_{M0}&\cdots&&&-B_{MM}
+
-B_{M0}&\cdots&-B_{MM}
 
\end{bmatrix}
 
\end{bmatrix}
 
\\
 
\\
Line 251: Line 171:
 
&
 
&
 
\begin{bmatrix}
 
\begin{bmatrix}
\ddots&&0\\
+
(k_0 + \kappa_0) \, B_{00}&\cdots&(k_M + \kappa_{0}) \, B_{0M}\\
&(k_l + \kappa_{n}) \, B_{nl}&\\
+
&&\\
0&&\ddots
+
\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}
 
\end{bmatrix}
 
\end{bmatrix}
Line 261: Line 183:
 
\vdots \\
 
\vdots \\
 
a_M \\
 
a_M \\
 +
\\
 
b_{0}\\
 
b_{0}\\
 
b_1 \\
 
b_1 \\
Line 274: Line 197:
 
\\
 
\\
 
0 \\
 
0 \\
 +
\\
 
2k_{0}A_{0} \\
 
2k_{0}A_{0} \\
 
0 \\
 
0 \\
Line 284: Line 208:
 
We then simply need to solve the linear system of equations.
 
We then simply need to solve the linear system of equations.
  
= Solution with Waves Incident at an Angle =
+
== 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
+
We can consider the problem when the waves are incident at an angle <math>\theta</math>.  
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).  
 
  
This means that the potential is now of the form <math>\phi(x,y,z)=e^{k_y y}\phi(x,z)</math>
+
{{incident angle}}
so that when we separate variables we obtain
 
  
 
Therefore the potential can
 
Therefore the potential can
Line 297: Line 218:
 
<center>
 
<center>
 
<math>
 
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x<0
+
\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>
 
</math>
 
</center>
 
</center>
Line 307: Line 228:
 
</math>
 
</math>
 
</center>
 
</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 <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.  
 
where we always take the positive real root or the root with positive imaginary part.  
  
Line 324: Line 245:
 
</math>
 
</math>
 
</center>
 
</center>
and these are solved exactly as before.  
+
and these are solved exactly as before.
  
= Matlab Code =
+
== Matlab Code ==
  
 
A program to calculate the coefficients for the semi-infinite dock problems can be found here
 
A program to calculate the coefficients for the semi-infinite dock problems can be found here
[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/semiinfinite_dock.m semiinfinite_dock.m]
+
{{semiinfinite_dock code}}
  
== Additional code ==
+
=== Additional code ===
  
 
This program requires
 
This program requires
[http://www.math.auckland.ac.nz/~meylan/code/dispersion/dispersion_free_surface.m dispersion_free_surface.m]
+
* {{free surface dispersion equation code}}
to run
 
  
 
[[Category:Eigenfunction Matching Method]]
 
[[Category:Eigenfunction Matching Method]]
[[Category:Pages with Matlab Code]]
 

Latest revision as of 00:43, 25 April 2017


Introduction

This is one of the simplest problem in eigenfunction matching. It also is an easy problem to understand the Wiener-Hopf and 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 Submerged Semi-Infinite Dock

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\gt 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

[math] \Delta\phi=0, \,\, -h \lt z \lt 0, [/math]

[math] \partial_z\phi=0, \,\, z=-h, [/math]

[math] \partial_z\phi=\alpha\phi, \,\, z=0,\,x\lt 0, [/math]

[math] \partial_z\phi=0, \,\, z=0,\,x\gt 0, [/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

We use separation of variables in the two regions, [math]x\lt 0[/math] and [math]x\gt 0[/math].

We express the potential as

[math] \phi(x,z) = X(x)Z(z)\, [/math]

and then Laplace's equation becomes

[math] \frac{X^{\prime\prime}}{X} = - \frac{Z^{\prime\prime}}{Z} = k^2 [/math]

Separation of variables for a free surface

We use separation of variables

We express the potential as

[math] \phi(x,z) = X(x)Z(z)\, [/math]

and then Laplace's equation becomes

[math] \frac{X^{\prime\prime}}{X} = - \frac{Z^{\prime\prime}}{Z} = k^2 [/math]

The separation of variables equation for deriving free surface eigenfunctions is as follows:

[math] Z^{\prime\prime} + k^2 Z =0. [/math]

subject to the boundary conditions

[math] Z^{\prime}(-h) = 0 [/math]

and

[math] Z^{\prime}(0) = \alpha Z(0) [/math]

We can then use the boundary condition at [math]z=-h \, [/math] to write

[math] Z = \frac{\cos k(z+h)}{\cos kh} [/math]

where we have chosen the value of the coefficent so we have unit value at [math]z=0[/math]. The boundary condition at the free surface ([math]z=0 \,[/math]) gives rise to:

[math] k\tan\left( kh\right) =-\alpha \, [/math]

which is the Dispersion Relation for a Free Surface

The above equation is a transcendental equation. If we solve for all roots in the complex plane we find that the first root is a pair of imaginary roots. We denote the imaginary solutions of this equation by [math]k_{0}=\pm ik \,[/math] and the positive real solutions by [math]k_{m} \,[/math], [math]m\geq1[/math]. The [math]k \,[/math] of the imaginary solution is the wavenumber. We put the imaginary roots back into the equation above and use the hyperbolic relations

[math] \cos ix = \cosh x, \quad \sin ix = i\sinh x, [/math]

to arrive at the dispersion relation

[math] \alpha = k\tanh kh. [/math]

We note that for a specified frequency [math]\omega \,[/math] the equation determines the wavenumber [math]k \,[/math].

Finally we define the function [math]Z(z) \,[/math] as

[math] \chi_{m}\left( z\right) =\frac{\cos k_{m}(z+h)}{\cos k_{m}h},\quad m\geq0 [/math]

as the vertical eigenfunction of the potential in the open water region. From Sturm-Liouville theory the vertical eigenfunctions are orthogonal. They can be normalised to be orthonormal, but this has no advantages for a numerical implementation. It can be shown that

[math] \int\nolimits_{-h}^{0}\chi_{m}(z)\chi_{n}(z) \mathrm{d} z=A_{n}\delta_{mn} [/math]

where

[math] A_{n}=\frac{1}{2}\left( \frac{\cos k_{n}h\sin k_{n}h+k_{n}h}{k_{n}\cos ^{2}k_{n}h}\right). [/math]


Separation of Variables for a Dock

The separation of variables equation for a floating dock

[math] Z^{\prime\prime} + k^2 Z =0, [/math]

subject to the boundary conditions

[math] Z^{\prime} (-h) = 0, [/math]

and

[math] Z^{\prime} (0) = 0. [/math]

The solution is [math]k=\kappa_{m}= \frac{m\pi}{h} \,[/math], [math]m\geq 0[/math] and

[math] Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad m\geq 0. [/math]

We note that

[math] \int\nolimits_{-h}^{0}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn}, [/math]

where

[math] C_{m} = \begin{cases} h,\quad m=0 \\ \frac{1}{2}h,\,\,\,m\neq 0 \end{cases} [/math]

Inner product between free surface and dock modes

[math] \int\nolimits_{-h}^{0}\phi_{n}(z)\psi_{m}(z) \mathrm{d} z=B_{mn} [/math]

where

[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\right) \left( k_{n} ^{2}-\kappa_{m}^{2}\right) } [/math]

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

[math] \phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x\lt 0 [/math]

and

[math] \phi(x,z)=\sum_{m=0}^{\infty}b_{m} e^{-\kappa_{m}x}\psi_{m}(z), \;\;x\gt 0 [/math]

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

To create meaningful solutions of the velocity potential [math]\phi[/math] in the specified domains we add an incident wave term to the expansion for the domain of [math]x \lt 0[/math] above. The incident potential is a wave of amplitude [math]A[/math] in displacement travelling in the positive [math]x[/math]-direction. We would only see this in the time domain [math]\Phi(x,z,t)[/math] however, in the frequency domain the incident potential can be written as

[math] \phi_{\mathrm{I}}(x,z) =e^{-k_{0}x}\chi_{0}\left( z\right). [/math]

The total velocity (scattered) potential now becomes [math]\phi = \phi_{\mathrm{I}} + \phi_{\mathrm{D}}[/math] for the domain of [math]x \lt 0[/math].

The first term in the expansion of the diffracted potential for the domain [math]x \lt 0[/math] is given by

[math] a_{0}e^{k_{0}x}\chi_{0}\left( z\right) [/math]

which represents the reflected wave.

In any scattering problem [math]|R|^2 + |T|^2 = 1[/math] where [math]R[/math] and [math]T[/math] are the reflection and transmission coefficients respectively. In our case of the semi-infinite dock [math]|a_{0}| = |R| = 1[/math] and [math]|T| = 0[/math] as there are no transmitted waves in the region under the dock.

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

[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]

and

[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]

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:

[math] A_{0}\delta_{0l}+a_{l}A_{l} =\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3) [/math]

and

[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]

If we mutiply equation (3) by [math]k_l \,[/math] and subtract equation (4) we obtain

[math] (k_{0}+k_l)A_{0}\delta_{0l} =\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(5) [/math]

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,

[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]

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].

When a wave in incident at an angle [math]\theta [/math] 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).

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

[math] k^2 = k_x^2 + k_y^2 [/math]

where [math]k[/math] is the separation constant calculated without an incident angle.

Therefore the potential can be expanded as

[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\lt 0 [/math]

and

[math] \phi(x,z)=\sum_{m=0}^{\infty}b_{m} e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x\gt 0 [/math]

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

[math] A_{0}\delta_{0l}+a_{l}A_{l} =\sum_{n=0}^{\infty}b_{m}B_{ml} [/math]

and

[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]

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.m

Additional code

This program requires