Difference between revisions of "Eigenfunction Matching for a Finite Dock"

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= Introduction =
+
{{complete pages}}
 +
 
 +
== Introduction ==
  
 
The problems consists of a region to the left and right
 
The problems consists of a region to the left and right
Line 5: Line 7:
 
not flow is possible.  
 
not flow is possible.  
 
We begin with the simple problem when the waves are normally incident (so that
 
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
+
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.
 
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
 
The theory is based on [[Eigenfunction Matching for a Semi-Infinite Dock]] and this should
be consulted for many details.
+
be consulted for many details. The solution here can be straightforwardly extended
 +
using [[:Category:Symmetry in Two Dimensions|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 [[:Category:Symmetry in Two Dimensions|Symmetry in Two Dimensions]] for the finite dock
 +
in [[Eigenfunction Matching for a Finite Dock using Symmetry]].
  
=Governing Equations=
+
==Governing Equations==
  
We begin with the [[Frequency Domain Problem]] for a dock which occupies
+
{{finite dock equations}}
the region <math>x>0</math>.
 
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=
+
==Solution Method==
  
 
We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the three regions, exactly
 
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]].  
 
as for the [[Eigenfunction Matching for a Semi-Infinite Dock]].  
 +
 +
{{separation of variables for a free surface}}
 +
 +
{{separation of variables for a dock}}
 +
 +
{{free surface dock relations}}
 +
 +
{{incident potential for two dimensions}}
 +
 +
== Expansion of the Potential ==
 
The potential can
 
The potential can
 
be expanded as
 
be expanded as
Line 56: Line 44:
 
<center>
 
<center>
 
<math>
 
<math>
\phi(x,z)=b_0 \frac{x+L}{2L}\psi_{0}(z) + \sum_{m=1}^{\infty}b_{m}
+
\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)
 
e^{-\kappa_{m} (x+L)}\psi_{m}(z)
+c_0 \frac{L-x}{2L}\psi_{0}(z)  + \sum_{m=1}^{\infty}c_{m}
+
+c_0 \frac{L+x}{2L}\psi_{0}(z)  + \sum_{m=1}^{\infty}c_{m}
 
e^{\kappa_{m} (x-L)}\psi_{m}(z)
 
e^{\kappa_{m} (x-L)}\psi_{m}(z)
 
, \;\;-L<x<L
 
, \;\;-L<x<L
Line 70: Line 58:
 
</math>
 
</math>
 
</center>
 
</center>
where <math>a_{m}</math> and <math>c_{m}</math>
+
where <math>a_{m}</math> and <math>d_{m}</math>
 
are the coefficients of the potential in the open water regions to the  
 
are the coefficients of the potential in the open water regions to the  
left and right and <math>b_m</math> are the coefficients under the ice
+
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.  
 
covered region. We have an incident wave from the left.  
the plate covered region respectively.
 
<math>k_n</math> are the roots of the
 
[[Dispersion Relation for a Free Surface]]. 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>.
 
<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==
+
===An infinite dimensional system of equations===
  
 
The potential and its derivative must be continuous across the
 
The potential and its derivative must be continuous across the
Line 141: Line 73:
 
\phi_{0}\left(  z\right) + \sum_{m=0}^{\infty}
 
\phi_{0}\left(  z\right) + \sum_{m=0}^{\infty}
 
a_{m} \phi_{m}\left(  z\right)  
 
a_{m} \phi_{m}\left(  z\right)  
=\sum_{m=1}^{\infty}b_{m}\psi_{m}(z) + \sum_{m=0}^{\infty}c_{m}\psi_{m}(z)e^{-2L\kappa_m}
+
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z) + \sum_{m=1}^{\infty}c_{m}\psi_{m}(z)e^{-2L\kappa_m}
 
</math>
 
</math>
 
</center>
 
</center>
Line 148: Line 80:
 
  -k_{0}\phi_{0}\left(  z\right)  +\sum
 
  -k_{0}\phi_{0}\left(  z\right)  +\sum
 
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left(  z\right)  
 
_{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
+
  =-\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) + \frac{c_0}{2L}\psi_0(z)+\sum_{m=1}^{\infty}c_{m}\kappa_{m}\psi
 
_{m}(z)e^{-2L\kappa_m}
 
_{m}(z)e^{-2L\kappa_m}
 
</math>
 
</math>
Line 155: Line 87:
 
<center>
 
<center>
 
<math>
 
<math>
\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)e^{-2L\kappa_m} + \sum_{m=1}^{\infty}c_{m}\psi_{m}(z)
+
\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)  
 
=\sum_{m=0}^{\infty}d_{m} \phi_{m}\left(  z\right)  
 
</math>
 
</math>
Line 161: Line 93:
 
<center>
 
<center>
 
<math>
 
<math>
\frac{b_0}{2L}\psi_0(z) -\sum_{m=1}^{\infty}b_{m}\kappa_{m}\psi
+
-\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)e^{-2L\kappa_m} + \frac{c_0}{2L}\psi_0(z)+\sum_{m=1}^{\infty}c_{m}\kappa_{m}\psi
 
_{m}(z)
 
_{m}(z)
 
= -\sum_{m=0}^{\infty}d_{m} k_m\phi_{m}\left(  z\right)  
 
= -\sum_{m=0}^{\infty}d_{m} k_m\phi_{m}\left(  z\right)  
 
</math>
 
</math>
 
</center>
 
</center>
for each <math>n</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:
Line 173: Line 105:
 
<math>
 
<math>
 
A_{0}\delta_{0l}+a_{l}A_{l}
 
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{m=1}^{\infty}b_{m}B_{ml}
+
=\sum_{m=0}^{\infty}b_{m}B_{ml}
+ \sum_{m=0}^{\infty}c_{m}B_{ml}e^{-2L\kappa_m}
+
+ \sum_{m=1}^{\infty}c_{m}B_{ml}e^{-2L\kappa_m}
 
</math>
 
</math>
 
</center>
 
</center>
Line 180: Line 112:
 
<math>
 
<math>
 
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
 
-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}  
+
  = - 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}
+
+ c_0 \frac{B_{0l}}{2L} + \sum_{m=1}^{\infty}c_{m}\kappa_{m}B_{ml} e^{-2L\kappa_m}
 
</math>
 
</math>
 
</center>
 
</center>
 
<center>
 
<center>
 
<math>
 
<math>
\sum_{m=0}^{\infty}b_{m}B_{ml}e^{-2L\kappa_m}
+
\sum_{m=1}^{\infty}b_{m}B_{ml}e^{-2L\kappa_m}
+ \sum_{m=1}^{\infty}c_{m}B_{ml}
+
+ \sum_{m=0}^{\infty}c_{m}B_{ml}
 
=d_l A_l
 
=d_l A_l
 
</math>
 
</math>
Line 193: Line 125:
 
<center>
 
<center>
 
<math>
 
<math>
b_0 \frac{B_{0l}}{2L} - \sum_{m=1}^{\infty}b_{m}\kappa_{m}B_{ml} e^{-2L\kappa_m}
+
- 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}  
+
+ c_0 \frac{B_{0l}}{2L} + \sum_{m=1}^{\infty}c_{m}\kappa_{m}B_{ml}  
 
= -d_l k_l A_l
 
= -d_l k_l A_l
 
</math>
 
</math>
 
</center>
 
</center>
  
=Numerical Solution=
+
==Numerical Solution==
  
 
To solve the system of equations we set the upper limit of <math>l</math> to
 
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.
 
be <math>M</math>. 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
 
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>
+
We can consider the problem when the waves are incident at an angle <math>\theta</math>.
so that when we separate variables we obtain
+
{{incident angle}}
  
 
Therefore the potential can
 
Therefore the potential can
Line 273: Line 199:
 
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 finite dock problems can be found here
 
A program to calculate the coefficients for the finite dock problems can be found here
[http://www.math.auckland.ac.nz/~meylan/code/finite_dock.m finite_dock.m]
+
[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/finite_dock.m finite_dock.m]
  
== Additional code ==
+
===Additional code ===
  
 
This program requires
 
This program requires
[http://www.math.auckland.ac.nz/~meylan/code/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]]
 
[[Category:Pages with Matlab Code]]

Latest revision as of 02:11, 7 April 2010


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 in Eigenfunction Matching for a Finite Dock using Symmetry.

Governing Equations

Wave scattering by a finite dock

We consider here the Frequency Domain Problem for a finite dock which occupies the region [math]\displaystyle{ -L\lt x\lt L }[/math] (we assume [math]\displaystyle{ e^{i\omega t} }[/math] time dependence). The water is assumed to have constant finite depth [math]\displaystyle{ h }[/math] and the [math]\displaystyle{ z }[/math]-direction points vertically upward with the water surface at [math]\displaystyle{ z=0 }[/math] and the sea floor at [math]\displaystyle{ z=-h }[/math]. The boundary value problem can therefore be expressed as

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

[math]\displaystyle{ \phi_{z}=0, \,\, z=-h, }[/math]

[math]\displaystyle{ \partial_z\phi=\alpha\phi, \,\, z=0,\,x \lt -L, {\rm or} \, x\gt L }[/math]

[math]\displaystyle{ \partial_z\phi=0, \,\, z=0,\,-L\lt x\lt L, }[/math]

We must also apply the Sommerfeld Radiation Condition as [math]\displaystyle{ |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 three regions, exactly as for the Eigenfunction Matching for a Semi-Infinite Dock.

Separation of variables for a free surface

We use separation of variables

We express the potential as

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

and then Laplace's equation becomes

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

subject to the boundary conditions

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

and

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

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

[math]\displaystyle{ 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]\displaystyle{ z=0 }[/math]. The boundary condition at the free surface ([math]\displaystyle{ z=0 \, }[/math]) gives rise to:

[math]\displaystyle{ 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]\displaystyle{ k_{0}=\pm ik \, }[/math] and the positive real solutions by [math]\displaystyle{ k_{m} \, }[/math], [math]\displaystyle{ m\geq1 }[/math]. The [math]\displaystyle{ 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]\displaystyle{ \cos ix = \cosh x, \quad \sin ix = i\sinh x, }[/math]

to arrive at the dispersion relation

[math]\displaystyle{ \alpha = k\tanh kh. }[/math]

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

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

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

where

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

subject to the boundary conditions

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

and

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

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

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

We note that

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

where

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

where

[math]\displaystyle{ 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]

Incident potential

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

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

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

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

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

which represents the reflected wave.

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

Expansion of the Potential

The potential can be expanded as

[math]\displaystyle{ \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\lt -L }[/math]

[math]\displaystyle{ \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\lt x\lt L }[/math]

and

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

where [math]\displaystyle{ a_{m} }[/math] and [math]\displaystyle{ d_{m} }[/math] are the coefficients of the potential in the open water regions to the left and right and [math]\displaystyle{ c_m }[/math] and [math]\displaystyle{ d_m }[/math] are the coefficients under the dock covered region. We have an incident wave from the left.

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]\displaystyle{ x=\pm L }[/math] have to be equal. We obtain

[math]\displaystyle{ \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]

[math]\displaystyle{ -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]

[math]\displaystyle{ \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]

[math]\displaystyle{ -\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]

We solve these equations by multiplying both equations by [math]\displaystyle{ \phi_{l}(z) }[/math] and integrating from [math]\displaystyle{ -h }[/math] to [math]\displaystyle{ 0 }[/math] to obtain:

[math]\displaystyle{ 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]

[math]\displaystyle{ -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]

[math]\displaystyle{ \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]

[math]\displaystyle{ - 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]

Numerical Solution

To solve the system of equations we set the upper limit of [math]\displaystyle{ l }[/math] to be [math]\displaystyle{ 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]\displaystyle{ \theta }[/math]. When a wave in incident at an angle [math]\displaystyle{ \theta }[/math] we have the wavenumber in the [math]\displaystyle{ y }[/math] direction is [math]\displaystyle{ k_y = \sin\theta k_0 }[/math] where [math]\displaystyle{ k_0 }[/math] is as defined previously (note that [math]\displaystyle{ k_y }[/math] is imaginary).

This means that the potential is now of the form [math]\displaystyle{ \phi(x,y,z)=e^{k_y y}\phi(x,z) }[/math] so that when we separate variables we obtain

[math]\displaystyle{ k^2 = k_x^2 + k_y^2 }[/math]

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

Therefore the potential can be expanded as

[math]\displaystyle{ \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\lt -L }[/math]

[math]\displaystyle{ \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\lt x\lt L }[/math]

and

[math]\displaystyle{ \phi(x,z)=\sum_{m=0}^{\infty}d_{m} e^{-\hat{k}_{m}(x-L)}\phi_{m}(z), \;\;L\lt x }[/math]

where [math]\displaystyle{ \hat{k}_{m} = \sqrt{k_m^2 + k_y^2} }[/math] and [math]\displaystyle{ \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]\displaystyle{ 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]

[math]\displaystyle{ -\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]

[math]\displaystyle{ \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]

[math]\displaystyle{ -\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]

and these are solved exactly as before.

Matlab Code

A program to calculate the coefficients for the finite dock problems can be found here finite_dock.m

Additional code

This program requires