WikiWaves - User contributions [en]

File:Thick inf dock.jpg

2008-06-26T12:13:13Z

F. Bonnefoy:

Eigenfunction Matching for a Submerged Semi-Infinite Dock

2008-06-26T09:35:57Z

F. Bonnefoy: /* Introduction */

= 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]]

[[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. Details can be found in [[Eigenfunction Matching for a Semi-Infinite Dock]].

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>

The potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-k_{0}^h x}\phi_{0}^h\left(
z\right) + \sum_{m=0}^{\infty}a_{m}e^{k_{m}^h x}\phi_{m}^h(z), \;\;x<0
</math>
</center>
<center>
<math>
\phi(x,z)= \sum_{m=0}^{\infty}b_{m}
e^{-k_{m}^d (x)}\phi_{m}^d(z)
, \;\;-d<z<0,\,\,x>0
</math>
</center>
and
<center>
<math>
\phi(x,z)= \sum_{m=0}^{\infty}c_{m}
e^{\kappa_{m} x}\psi_{m}(z)
, \;\;-h<z<-d,\,\,x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water regions to the
left and right and <math>c_m</math> are the coefficients under the dock
covered region. We have an incident wave from the left.
<math>k_n^l</math> are the roots of the
[[Dispersion Relation for a Free Surface]]
<center>
<math> k \tan(kl) = -\alpha</math>
</center>We denote the
positive imaginary solutions by <math>k_{0}^l</math> and
the positive real solutions by <math>k_{m}^l</math>, <math>m\geq1</math> (ordered with increasing
imaginary part) and
<math>\kappa_{m}=m\pi/(h-d)</math>. We define
<center>
<math>
\phi_{m}^l\left( z\right) = \frac{\cos k_{m}(z+l)}{\cos k_{m}l},\quad m\geq0
</math>
</center>
as the vertical eigenfunction of the potential in the open
water regions 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. We define
<center>
<math>
\int\nolimits_{-d}^{0}\phi_{m}^d(z)\phi_{n}^d(z) d z=A_{m}\delta_{mn}
</math>
</center>
where
<center>
<math>
A_{m}=\frac{1}{2}\left( \frac{\cos k_{m}d\sin k_{m}d+k_{m}d}{k_{m}\cos
^{2}k_{m}l}\right)
</math>
</center>
<center>
<math>
\int\nolimits_{-d}^{0}\phi_{n}^h(z)\phi_{m}^d(z) d z=B_{mn}
</math>
</center>
and
<center>
<math>
\int\nolimits_{-h}^{-d}\phi_{n}^h(z)\psi_{m}(z) d z=C_{mn}
</math>
</center>
and
<center>
<math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) d z=D_{m}\delta_{mn}
</math>
</center>
where
<center>
<math>
D_{m}=\frac{1}{2}(h-d),\quad,m\neq 0 \quad \mathrm{and} \quad D_0 = (h-d)
</math></center>

==An infinite dimensional system of equations==

The potential and its derivative must be continuous across the
transition from open water to the dock region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}^h\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\phi_{m}^d(z),\,\,\,-d<z<0
</math>
</center>
<center>
<math>
\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}^h\left( z\right)
=\sum_{m=0}^{\infty}c_{m}\psi_{m}(z),\,\,\,-h<z<-d
</math>
</center>
<center>
<math>
-k_0^h\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty}
k_m^h a_{m} \phi_{m}^h\left( z\right)
=\sum_{m=0}^{\infty}k_m^d b_{m}\phi_{m}^d(z),\,\,\,-d<z<0
</math>
</center>
<center>
<math>
-k_0^h\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty}
k_m^h a_{m} \phi_{m}^h\left( z\right)
=\sum_{m=0}^{\infty}\kappa_m c_{m}\psi_{m}(z),\,\,\,-h<z<-d
</math>
</center>

The standard method to solve these equations (from [[Linton and Evans]]) is to
mutiply both equations by
<math>\phi_{q}^d(z)</math> and integrating from <math>-d</math> to <math>0</math> or
by multiplying both equations by
<math>\psi_{r}(z)</math> and integrating from <math>-h</math> to <math>-d</math>.
However, we use a different method, which is closer to the solution method
for [[Eigenfunction Matching for a Semi-Infinite Dock]] which allows us to keep
the computer code similar. These is no significant difference between the methods
numerically and a close connection exists.

We truncate the sum to <math>N+1</math> modes and introduce a new function
<center>
<math>
\chi_m =
\begin{array}
\psi_{m}(z),\,\,\,-h<z<-d
</math>
</center>
We multiply each equation by <math>\phi_{q}^h(z)</math> and integrating
from <math>-h</math> to <math>0</math> to obtain

<center>
<math>
A_{0q} + \sum_{m=0}^{\infty}a_{q}A_{mq}
= b_qB_{q}
</math>
</center>
<center>
<math>
-k_{0}^h A_{0q} + \sum_{m=0}^{\infty}k_{m}^h a_{m}A_{mq}
= -k_{q}^d b_qB_{q}
</math>
</center>

<center>
<math>
C_{0r} + \sum_{m=0}^{\infty}a_{r}C_{mr}
= c_rD_{r}
</math>
</center>
<center>
<math>
-k_{0}^h C_{0r} + \sum_{m=0}^{\infty}k_{m}^h a_{r}C_{mr}
= -k_{r}^d c_rD_{r}
</math>
</center>

=Numerical Solution=

The numerical truncation needs to be handled with some care. We need the number of eigenfunctions to the left and right
to match. We choose the number of eigenfunctions to the left to be <math>M</math> and then choose
the number above and below to sum to <math>M</math> in proportion as <math>h</math> is to <math>d</math>
(making sure that there is at least one eigenfunction in each region.

= Solution with Waves Incident at an Angle =

We can consider the problem when the waves are incident at an angle <math>\theta</math> but this
is not presented here. For details see [[Eigenfunction Matching for a Semi-Infinite Dock]].

= 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
[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]]

File:Submerged inf dock.jpg

2008-06-26T09:35:07Z

F. Bonnefoy:

Eigenfunction Matching for a Submerged Semi-Infinite Dock

2008-06-26T09:09:01Z

F. Bonnefoy: /* Solution Method */

= 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]]

=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. Details can be found in [[Eigenfunction Matching for a Semi-Infinite Dock]].

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>

The potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-k_{0}^h x}\phi_{0}^h\left(
z\right) + \sum_{m=0}^{\infty}a_{m}e^{k_{m}^h x}\phi_{m}^h(z), \;\;x<0
</math>
</center>
<center>
<math>
\phi(x,z)= \sum_{m=0}^{\infty}b_{m}
e^{-k_{m}^d (x)}\phi_{m}^d(z)
, \;\;-d<z<0,\,\,x>0
</math>
</center>
and
<center>
<math>
\phi(x,z)= \sum_{m=0}^{\infty}c_{m}
e^{\kappa_{m} x}\psi_{m}(z)
, \;\;-h<z<-d,\,\,x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water regions to the
left and right and <math>c_m</math> are the coefficients under the dock
covered region. We have an incident wave from the left.
<math>k_n^l</math> are the roots of the
[[Dispersion Relation for a Free Surface]]
<center>
<math> k \tan(kl) = -\alpha</math>
</center>We denote the
positive imaginary solutions by <math>k_{0}^l</math> and
the positive real solutions by <math>k_{m}^l</math>, <math>m\geq1</math> (ordered with increasing
imaginary part) and
<math>\kappa_{m}=m\pi/(h-d)</math>. We define
<center>
<math>
\phi_{m}^l\left( z\right) = \frac{\cos k_{m}(z+l)}{\cos k_{m}l},\quad m\geq0
</math>
</center>
as the vertical eigenfunction of the potential in the open
water regions 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. We define
<center>
<math>
\int\nolimits_{-d}^{0}\phi_{m}^d(z)\phi_{n}^d(z) d z=A_{m}\delta_{mn}
</math>
</center>
where
<center>
<math>
A_{m}=\frac{1}{2}\left( \frac{\cos k_{m}d\sin k_{m}d+k_{m}d}{k_{m}\cos
^{2}k_{m}l}\right)
</math>
</center>
<center>
<math>
\int\nolimits_{-d}^{0}\phi_{n}^h(z)\phi_{m}^d(z) d z=B_{mn}
</math>
</center>
and
<center>
<math>
\int\nolimits_{-h}^{-d}\phi_{n}^h(z)\psi_{m}(z) d z=C_{mn}
</math>
</center>
and
<center>
<math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) d z=D_{m}\delta_{mn}
</math>
</center>
where
<center>
<math>
D_{m}=\frac{1}{2}(h-d),\quad,m\neq 0 \quad \mathrm{and} \quad D_0 = (h-d)
</math></center>

==An infinite dimensional system of equations==

The potential and its derivative must be continuous across the
transition from open water to the dock region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}^h\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\phi_{m}^d(z),\,\,\,-d<z<0
</math>
</center>
<center>
<math>
\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}^h\left( z\right)
=\sum_{m=0}^{\infty}c_{m}\psi_{m}(z),\,\,\,-h<z<-d
</math>
</center>
<center>
<math>
-k_0^h\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty}
k_m^h a_{m} \phi_{m}^h\left( z\right)
=\sum_{m=0}^{\infty}k_m^d b_{m}\phi_{m}^d(z),\,\,\,-d<z<0
</math>
</center>
<center>
<math>
-k_0^h\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty}
k_m^h a_{m} \phi_{m}^h\left( z\right)
=\sum_{m=0}^{\infty}\kappa_m c_{m}\psi_{m}(z),\,\,\,-h<z<-d
</math>
</center>

The standard method to solve these equations (from [[Linton and Evans]]) is to
mutiply both equations by
<math>\phi_{q}^d(z)</math> and integrating from <math>-d</math> to <math>0</math> or
by multiplying both equations by
<math>\psi_{r}(z)</math> and integrating from <math>-h</math> to <math>-d</math>.
However, we use a different method, which is closer to the solution method
for [[Eigenfunction Matching for a Semi-Infinite Dock]] which allows us to keep
the computer code similar. These is no significant difference between the methods
numerically and a close connection exists.

We truncate the sum to <math>N+1</math> modes and introduce a new function
<center>
<math>
\chi_m =
\begin{array}
\psi_{m}(z),\,\,\,-h<z<-d
</math>
</center>
We multiply each equation by <math>\phi_{q}^h(z)</math> and integrating
from <math>-h</math> to <math>0</math> to obtain

<center>
<math>
A_{0q} + \sum_{m=0}^{\infty}a_{q}A_{mq}
= b_qB_{q}
</math>
</center>
<center>
<math>
-k_{0}^h A_{0q} + \sum_{m=0}^{\infty}k_{m}^h a_{m}A_{mq}
= -k_{q}^d b_qB_{q}
</math>
</center>

<center>
<math>
C_{0r} + \sum_{m=0}^{\infty}a_{r}C_{mr}
= c_rD_{r}
</math>
</center>
<center>
<math>
-k_{0}^h C_{0r} + \sum_{m=0}^{\infty}k_{m}^h a_{r}C_{mr}
= -k_{r}^d c_rD_{r}
</math>
</center>

=Numerical Solution=

The numerical truncation needs to be handled with some care. We need the number of eigenfunctions to the left and right
to match. We choose the number of eigenfunctions to the left to be <math>M</math> and then choose
the number above and below to sum to <math>M</math> in proportion as <math>h</math> is to <math>d</math>
(making sure that there is at least one eigenfunction in each region.

= Solution with Waves Incident at an Angle =

We can consider the problem when the waves are incident at an angle <math>\theta</math> but this
is not presented here. For details see [[Eigenfunction Matching for a Semi-Infinite Dock]].

= 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
[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]]

Eigenfunction Matching for a Submerged Semi-Infinite Dock

2008-06-26T09:08:47Z

F. Bonnefoy: /* Governing Equations */

= 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]]

=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. Details can be found in [[Eigenfunction Matching for a Semi-Infinite Dock]].

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>

The potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-k_{0}^h x}\phi_{0}^h\left(
z\right) + \sum_{m=0}^{\infty}a_{m}e^{k_{m}^h x}\phi_{m}^h(z), \;\;x<0
</math>
</center>
<center>
<math>
\phi(x,z)= \sum_{m=0}^{\infty}b_{m}
e^{-k_{m}^d (x)}\phi_{m}^d(z)
, \;\;-d<z<0,\,\,x>0
</math>
</center>
and
<center>
<math>
\phi(x,z)= \sum_{m=0}^{\infty}c_{m}
e^{\kappa_{m} x}\psi_{m}(z)
, \;\;-h<z<-d,\,\,x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water regions to the
left and right and <math>c_m</math> are the coefficients under the dock
covered region. We have an incident wave from the left.
<math>k_n^l</math> are the roots of the
[[Dispersion Relation for a Free Surface]]
<center>
<math> k \tan(kl) = -\alpha</math>
</center>We denote the
positive imaginary solutions by <math>k_{0}^l</math> and
the positive real solutions by <math>k_{m}^l</math>, <math>m\geq1</math> (ordered with increasing
imaginary part) and
<math>\kappa_{m}=m\pi/(h-d)</math>. We define
<center>
<math>
\phi_{m}^l\left( z\right) = \frac{\cos k_{m}(z+l)}{\cos k_{m}l},\quad m\geq0
</math>
</center>
as the vertical eigenfunction of the potential in the open
water regions 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. We define
<center>
<math>
\int\nolimits_{-d}^{0}\phi_{m}^d(z)\phi_{n}^d(z) d z=A_{m}\delta_{mn}
</math>
</center>
where
<center>
<math>
A_{m}=\frac{1}{2}\left( \frac{\cos k_{m}d\sin k_{m}d+k_{m}d}{k_{m}\cos
^{2}k_{m}l}\right)
</math>
</center>
<center>
<math>
\int\nolimits_{-d}^{0}\phi_{n}^h(z)\phi_{m}^d(z) d z=B_{mn}
</math>
</center>
and
<center>
<math>
\int\nolimits_{-h}^{-d}\phi_{n}^h(z)\psi_{m}(z) d z=C_{mn}
</math>
</center>
and
<center>
<math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) d z=D_{m}\delta_{mn}
</math>
</center>
where
<center>
<math>
D_{m}=\frac{1}{2}(h-d),\quad,m\neq 0 \quad \mathrm{and} \quad D_0 = (h-d)
</math></center>

==An infinite dimensional system of equations==

The potential and its derivative must be continuous across the
transition from open water to the dock region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}^h\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\phi_{m}^d(z),\,\,\,-d<z<0
</math>
</center>
<center>
<math>
\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}^h\left( z\right)
=\sum_{m=0}^{\infty}c_{m}\psi_{m}(z),\,\,\,-h<z<-d
</math>
</center>
<center>
<math>
-k_0^h\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty}
k_m^h a_{m} \phi_{m}^h\left( z\right)
=\sum_{m=0}^{\infty}k_m^d b_{m}\phi_{m}^d(z),\,\,\,-d<z<0
</math>
</center>
<center>
<math>
-k_0^h\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty}
k_m^h a_{m} \phi_{m}^h\left( z\right)
=\sum_{m=0}^{\infty}\kappa_m c_{m}\psi_{m}(z),\,\,\,-h<z<-d
</math>
</center>

The standard method to solve these equations (from [[Linton and Evans]]) is to
mutiply both equations by
<math>\phi_{q}^d(z)</math> and integrating from <math>-d</math> to <math>0</math> or
by multiplying both equations by
<math>\psi_{r}(z)</math> and integrating from <math>-h</math> to <math>-d</math>.
However, we use a different method, which is closer to the solution method
for [[Eigenfunction Matching for a Semi-Infinite Dock]] which allows us to keep
the computer code similar. These is no significant difference between the methods
numerically and a close connection exists.

We truncate the sum to <math>N+1</math> modes and introduce a new function
<center>
<math>
\chi_m =
\begin{array}
\psi_{m}(z),\,\,\,-h<z<-d
</math>
</center>
We multiply each equation by <math>\phi_{q}^h(z)</math> and integrating
from <math>-h</math> to <math>0</math> to obtain

<center>
<math>
A_{0q} + \sum_{m=0}^{\infty}a_{q}A_{mq}
= b_qB_{q}
</math>
</center>
<center>
<math>
-k_{0}^h A_{0q} + \sum_{m=0}^{\infty}k_{m}^h a_{m}A_{mq}
= -k_{q}^d b_qB_{q}
</math>
</center>

<center>
<math>
C_{0r} + \sum_{m=0}^{\infty}a_{r}C_{mr}
= c_rD_{r}
</math>
</center>
<center>
<math>
-k_{0}^h C_{0r} + \sum_{m=0}^{\infty}k_{m}^h a_{r}C_{mr}
= -k_{r}^d c_rD_{r}
</math>
</center>

=Numerical Solution=

The numerical truncation needs to be handled with some care. We need the number of eigenfunctions to the left and right
to match. We choose the number of eigenfunctions to the left to be <math>M</math> and then choose
the number above and below to sum to <math>M</math> in proportion as <math>h</math> is to <math>d</math>
(making sure that there is at least one eigenfunction in each region.

= Solution with Waves Incident at an Angle =

We can consider the problem when the waves are incident at an angle <math>\theta</math> but this
is not presented here. For details see [[Eigenfunction Matching for a Semi-Infinite Dock]].

= 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
[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]]

Eigenfunction Matching for a Submerged Semi-Infinite Dock

2008-06-26T09:07:28Z

F. Bonnefoy: /* Introduction */

= 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]]

=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. Details can be found in [[Eigenfunction Matching for a Semi-Infinite Dock]].

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>

The potential can
be expanded as
<center>
<math>
\phi(x,z)=e^{-k_{0}^h x}\phi_{0}^h\left(
z\right) + \sum_{m=0}^{\infty}a_{m}e^{k_{m}^h x}\phi_{m}^h(z), \;\;x<0
</math>
</center>
<center>
<math>
\phi(x,z)= \sum_{m=0}^{\infty}b_{m}
e^{-k_{m}^d (x)}\phi_{m}^d(z)
, \;\;-d<z<0,\,\,x>0
</math>
</center>
and
<center>
<math>
\phi(x,z)= \sum_{m=0}^{\infty}c_{m}
e^{\kappa_{m} x}\psi_{m}(z)
, \;\;-h<z<-d,\,\,x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water regions to the
left and right and <math>c_m</math> are the coefficients under the dock
covered region. We have an incident wave from the left.
<math>k_n^l</math> are the roots of the
[[Dispersion Relation for a Free Surface]]
<center>
<math> k \tan(kl) = -\alpha</math>
</center>We denote the
positive imaginary solutions by <math>k_{0}^l</math> and
the positive real solutions by <math>k_{m}^l</math>, <math>m\geq1</math> (ordered with increasing
imaginary part) and
<math>\kappa_{m}=m\pi/(h-d)</math>. We define
<center>
<math>
\phi_{m}^l\left( z\right) = \frac{\cos k_{m}(z+l)}{\cos k_{m}l},\quad m\geq0
</math>
</center>
as the vertical eigenfunction of the potential in the open
water regions 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. We define
<center>
<math>
\int\nolimits_{-d}^{0}\phi_{m}^d(z)\phi_{n}^d(z) d z=A_{m}\delta_{mn}
</math>
</center>
where
<center>
<math>
A_{m}=\frac{1}{2}\left( \frac{\cos k_{m}d\sin k_{m}d+k_{m}d}{k_{m}\cos
^{2}k_{m}l}\right)
</math>
</center>
<center>
<math>
\int\nolimits_{-d}^{0}\phi_{n}^h(z)\phi_{m}^d(z) d z=B_{mn}
</math>
</center>
and
<center>
<math>
\int\nolimits_{-h}^{-d}\phi_{n}^h(z)\psi_{m}(z) d z=C_{mn}
</math>
</center>
and
<center>
<math>
\int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) d z=D_{m}\delta_{mn}
</math>
</center>
where
<center>
<math>
D_{m}=\frac{1}{2}(h-d),\quad,m\neq 0 \quad \mathrm{and} \quad D_0 = (h-d)
</math></center>

==An infinite dimensional system of equations==

The potential and its derivative must be continuous across the
transition from open water to the dock region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}^h\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\phi_{m}^d(z),\,\,\,-d<z<0
</math>
</center>
<center>
<math>
\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}^h\left( z\right)
=\sum_{m=0}^{\infty}c_{m}\psi_{m}(z),\,\,\,-h<z<-d
</math>
</center>
<center>
<math>
-k_0^h\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty}
k_m^h a_{m} \phi_{m}^h\left( z\right)
=\sum_{m=0}^{\infty}k_m^d b_{m}\phi_{m}^d(z),\,\,\,-d<z<0
</math>
</center>
<center>
<math>
-k_0^h\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty}
k_m^h a_{m} \phi_{m}^h\left( z\right)
=\sum_{m=0}^{\infty}\kappa_m c_{m}\psi_{m}(z),\,\,\,-h<z<-d
</math>
</center>

The standard method to solve these equations (from [[Linton and Evans]]) is to
mutiply both equations by
<math>\phi_{q}^d(z)</math> and integrating from <math>-d</math> to <math>0</math> or
by multiplying both equations by
<math>\psi_{r}(z)</math> and integrating from <math>-h</math> to <math>-d</math>.
However, we use a different method, which is closer to the solution method
for [[Eigenfunction Matching for a Semi-Infinite Dock]] which allows us to keep
the computer code similar. These is no significant difference between the methods
numerically and a close connection exists.

We truncate the sum to <math>N+1</math> modes and introduce a new function
<center>
<math>
\chi_m =
\begin{array}
\psi_{m}(z),\,\,\,-h<z<-d
</math>
</center>
We multiply each equation by <math>\phi_{q}^h(z)</math> and integrating
from <math>-h</math> to <math>0</math> to obtain

<center>
<math>
A_{0q} + \sum_{m=0}^{\infty}a_{q}A_{mq}
= b_qB_{q}
</math>
</center>
<center>
<math>
-k_{0}^h A_{0q} + \sum_{m=0}^{\infty}k_{m}^h a_{m}A_{mq}
= -k_{q}^d b_qB_{q}
</math>
</center>

<center>
<math>
C_{0r} + \sum_{m=0}^{\infty}a_{r}C_{mr}
= c_rD_{r}
</math>
</center>
<center>
<math>
-k_{0}^h C_{0r} + \sum_{m=0}^{\infty}k_{m}^h a_{r}C_{mr}
= -k_{r}^d c_rD_{r}
</math>
</center>

=Numerical Solution=

The numerical truncation needs to be handled with some care. We need the number of eigenfunctions to the left and right
to match. We choose the number of eigenfunctions to the left to be <math>M</math> and then choose
the number above and below to sum to <math>M</math> in proportion as <math>h</math> is to <math>d</math>
(making sure that there is at least one eigenfunction in each region.

= Solution with Waves Incident at an Angle =

We can consider the problem when the waves are incident at an angle <math>\theta</math> but this
is not presented here. For details see [[Eigenfunction Matching for a Semi-Infinite Dock]].

= 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
[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]]

Eigenfunction Matching for a Finite Dock

2008-06-25T20:11:26Z

F. Bonnefoy:

= 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{\cos\kappa_{m} x}{\cos \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}\tan\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
=-\sum_{m=0}^{\infty}b_{m}^{a}\kappa_{m}\cot\kappa_{m}L B_{ml}
</math>
</center>
where for <math>m=0</math>, we adopt the notation <math>\, \cot\kappa_{0}L=1/L</math>.

(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>

Eigenfunction Matching for a Finite Dock

2008-06-25T20:06:06Z

F. Bonnefoy: /* Solution to the original problem */

= 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{\cos\kappa_{m} x}{\cos \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}\tan\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
=-\sum_{m=0}^{\infty}b_{m}^{a}\kappa_{m}\cot\kappa_{m}L B_{ml}
</math>
</center>
where for <math>m=0</math>, we adopt the notation <math>\, \cot\kappa_{0}L=1/L</math>.

(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}
e^{-\kappa_{m} (x+L)}\psi_{m}(z)
-c_0 \frac{x}{L} \psi_{0}(z) + \sum_{m=1}^{\infty}c_{m}
e^{\kappa_{m} (x-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>

Eigenfunction Matching for a Finite Dock

2008-06-25T20:01:41Z

F. Bonnefoy: /* Solution to the original problem */

= 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{\cos\kappa_{m} x}{\cos \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}\tan\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
=-\sum_{m=0}^{\infty}b_{m}^{a}\kappa_{m}\cot\kappa_{m}L B_{ml}
</math>
</center>
where for <math>m=0</math>, we adopt the notation <math>\, \cot\kappa_{0}L=1/L</math>.

(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 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>

Eigenfunction Matching for a Finite Dock

2008-06-25T19:49:18Z

F. Bonnefoy: /* Anti-Symmetric solution */

= 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{\cos\kappa_{m} x}{\cos \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}\tan\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
=-\sum_{m=0}^{\infty}b_{m}^{a}\kappa_{m}\cot\kappa_{m}L B_{ml}
</math>
</center>
where for <math>m=0</math>, we adopt the notation <math>\, \cot\kappa_{0}L=1/L</math>.

(for full details of this derivation see [[Eigenfunction Matching for a Semi-Infinite Dock]])

== Solution to the original problem ==

The amplitude in the open-water region is simply obtained by the superposition principle
<math>a_{m}=\frac{1}{2}\left(a_{m}^{s}+a_{m}^{a}\right)</math>
and in the dock-covered region <math>b_{m}=\frac{1}{2}\left(b_{m}^{s}+b_{m}^{a}\right)</math>.

Eigenfunction Matching for a Finite Dock

2008-06-25T19:41:00Z

F. Bonnefoy: /* Solution using Symmetry */

= 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{\cos\kappa_{m} x}{\cos \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}\tan\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
=-\sum_{m=0}^{\infty}b_{m}^{a}\kappa_{m}\cot\kappa_{m}L B_{ml}
</math>
</center>
where for <math>m=0</math>, we adopt the notation <math>\, \cot\kappa_{0}L=1/L</math>.

(for full details of this derivation see [[Eigenfunction Matching for a Semi-Infinite Dock]])

== Solution to the original problem ==

The amplitude in the open-water region is simply obtained by the superposition principle
<math>a_{m}=\frac{1}{2}\left(a_{m}^{s}+a_{m}^{a}\right)</math>
and in the dock-covered region <math>b_{m}=\frac{1}{2}\left(b_{m}^{s}+b_{m}^{a}\right)</math>.

Two Identical Docks using Symmetry

2008-06-25T19:11:11Z

F. Bonnefoy: /* Introduction */

= Introduction =

The problems consists three regions
with a free surface and and two regions of identical length with a rigid surface through which
not flow is possible.
The solution method is an extension of [[Eigenfunction Matching for a Finite Dock]]
using [[Symmetry in Two Dimensions]].
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.

=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>
\phi_{z}=0, \,\, z=-h,
</math>
</center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<-L_2,\,-L_1<x<L_1, {\rm or} \, x>L_2
</math></center>
<center>
<math>
\partial_z\phi=0, \,\, z=0,\,-L_2<x<-L_1, {\rm or} \, L_1<x<L_2,
</math>
</center>
where we require <math>L_1<L_2</math> and we define <math>L_2 - L_1 = 2L</math>
(so that the dock also has length <math>2L</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=

The solution method uses [[Symmetry in Two Dimensions]] and we write the potential as
a symmetric and an anti-symmetric part and consider only the region <math>x<0</math>. We apply
either Neuman (symmetric) or Dirichlet (anti-symmetric) boundary conditions at <math>x=0</math>.
We [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the three regions, similar to
as for the [[Eigenfunction Matching for a Finite Dock]]. We being with the symmetric potential which can
be expanded as
<center>
<math>
\phi^s(x,z)=e^{-k_{0}(x+L_2)}\phi_{0}\left(
z\right) + \sum_{m=0}^{\infty}a^s_{m}e^{k_{m}(x+L_2)}\phi_{m}(z), \;\;x<-L_2
</math>
</center>
<center>
<math>
\phi(x,z)=b^s_0 \frac{x+L_1}{-2L}\psi_{0}(z) + \sum_{m=1}^{\infty}b^s_{m}
e^{-\kappa_{m} (x+L_2)}\psi_{m}(z)
+c^s_0 \frac{L_2+x}{2L}\psi_{0}(z) + \sum_{m=1}^{\infty}c^s_{m}
e^{\kappa_{m} (x+L_1)}\psi_{m}(z)
, \;\;-L_2<x<L_1
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}d^s_{m} \frac{\cosh(k_{m}x)}{\cosh(k_m L_1)} \phi_{m}(z), \;\;-L_1<x<0
</math>
</center>
where <math>a^s_{m}</math> and <math>d^s_{m}</math>
are the coefficients of the potential in the open water regions to the
left and right and <math>b^s_m</math> and <math>c^s_m</math> are the coefficients under the dock.
<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=- L_2</math> and
<math>x=-L_1</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a^s_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b^s_{m}\psi_{m}(z) + \sum_{m=1}^{\infty}c^s_{m}\psi_{m}(z)e^{-2L\kappa_m}
</math>
</center>
<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a^s_{m}k_{m}\phi_{m}\left( z\right)
=-\frac{b^s_0}{2L}\psi_0(z) -\sum_{m=1}^{\infty}b^s_{m}\kappa_{m}\psi
_{m}(z) + \frac{c^s_0}{2L}\psi_0(z)+\sum_{m=1}^{\infty}c^s_{m}\kappa_{m}\psi
_{m}(z)e^{-2L\kappa_m}
</math>
</center>
<center>
<math>
\sum_{m=1}^{\infty}b^s_{m}\psi_{m}(z)e^{-2L\kappa_m} + \sum_{m=0}^{\infty}c^s_{m}\psi_{m}(z)
=\sum_{m=0}^{\infty}d^s_{m} \phi_{m}\left( z\right)
</math>
</center>
<center>
<math>
-\frac{b^s_0}{2L}\psi_0(z) -\sum_{m=1}^{\infty}b^s_{m}\kappa_{m}\psi
_{m}(z)e^{-2L\kappa_m} + \frac{c^s_0}{2L}\psi_0(z)+\sum_{m=1}^{\infty}c^s_{m}\kappa_{m}\psi
_{m}(z)
= \sum_{m=0}^{\infty}d^s_{m} \tanh(k_m L_1) 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^s_{l}A_{l}
=\sum_{m=0}^{\infty}b^s_{m}B_{ml}
+ \sum_{m=1}^{\infty}c^s_{m}B_{ml}e^{-2L\kappa_m}
</math>
</center>
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a^s_{l}k_{l}A_l
= - b^s_0 \frac{B_{0l}}{2L} - \sum_{m=1}^{\infty}b^s_{m}\kappa_{m}B_{ml}
+ c^s_0 \frac{B_{0l}}{2L} + \sum_{m=1}^{\infty}c^s_{m}\kappa_{m}B_{ml} e^{-2L\kappa_m}
</math>
</center>
<center>
<math>
\sum_{m=1}^{\infty}b^s_{m}B_{ml}e^{-2L\kappa_m}
+ \sum_{m=0}^{\infty}c^s_{m}B_{ml}
=d^s_l A_l
</math>
</center>
<center>
<math>
- b^s_0 \frac{B_{0l}}{2L} - \sum_{m=1}^{\infty}b^s_{m}\kappa_{m}B_{ml} e^{-2L\kappa_m}
+ c^s_0 \frac{B_{0l}}{2L} + \sum_{m=1}^{\infty}c^s_{m}\kappa_{m}B_{ml}
= d^s_l \tanh(k_l L_1) 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.

= Anti-Symmetric Solution =

The solution for the anti-symmetric potential proceeds in an almost identical manner.

<center>
<math>
\phi^a(x,z)=e^{-k_{0}(x+L_2)}\phi_{0}\left(
z\right) + \sum_{m=0}^{\infty}a^a_{m}e^{k_{m}(x+L_2)}\phi_{m}(z), \;\;x<-L_2
</math>
</center>
<center>
<math>
\phi(x,z)=b^a_0 \frac{x+L_1}{-2L}\psi_{0}(z) + \sum_{m=1}^{\infty}b^a_{m}
e^{-\kappa_{m} (x+L_2)}\psi_{m}(z)
+c^a_0 \frac{L_2+x}{2L}\psi_{0}(z) + \sum_{m=1}^{\infty}c^a_{m}
e^{\kappa_{m} (x+L_1)}\psi_{m}(z)
, \;\;-L_2<x<L_1
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}d^a_{m} \frac{\sinh(k_{m}x)}{-\sinh(k_m L_1)} \phi_{m}(z), \;\;-L_1<x<0
</math>
</center>
where <math>a^a_{m}</math> and <math>d^a_{m}</math>
are the coefficients of the potential in the open water regions to the
left and right and <math>b^a_m</math> and <math>c^a_m</math> are the coefficients under the dock.

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^a_{l}A_{l}
=\sum_{m=0}^{\infty}b^a_{m}B_{ml}
+ \sum_{m=1}^{\infty}c^a_{m}B_{ml}e^{-2L\kappa_m}
</math>
</center>
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a^a_{l}k_{l}A_l
= - b^a_0 \frac{B_{0l}}{2L} - \sum_{m=1}^{\infty}b^a_{m}\kappa_{m}B_{ml}
+ c^a_0 \frac{B_{0l}}{2L} + \sum_{m=1}^{\infty}c^a_{m}\kappa_{m}B_{ml} e^{-2L\kappa_m}
</math>
</center>
<center>
<math>
\sum_{m=1}^{\infty}b^a_{m}B_{ml}e^{-2L\kappa_m}
+ \sum_{m=0}^{\infty}c^a_{m}B_{ml}
=d^a_l A_l
</math>
</center>
<center>
<math>
- b^a_0 \frac{B_{0l}}{2L} - \sum_{m=1}^{\infty}b^a_{m}\kappa_{m}B_{ml} e^{-2L\kappa_m}
+ c^a_0 \frac{B_{0l}}{2L} + \sum_{m=1}^{\infty}c^a_{m}\kappa_{m}B_{ml}
= d^a_l \frac{k_l A_l}{\tanh(k_l L_1)}
</math>
</center>

= 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>.
Therefore the symmetric potential can
be expanded as
<center>
<math>
\phi^{s}_{0}\left( z\right) + \sum_{m=0}^{\infty}
a^s_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b^s_{m}\psi_{m}(z) + \sum_{m=0}^{\infty}c^s_{m}\psi_{m}(z)e^{-2L\hat{\kappa}_m}
</math>
</center>
<center>
<math>
-\hat{k}_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a^s_{m}\hat{k}_{m}\phi_{m}\left( z\right)
= -\sum_{m=0}^{\infty}b^s_{m}\hat{\kappa}_{m}\psi
_{m}(z) +\sum_{m=0}^{\infty}c^s_{m}\hat{\kappa}_{m}\psi
_{m}(z)e^{-2L\hat{\kappa}_m}
</math>
</center>
<center>
<math>
\sum_{m=1}^{\infty}b^s_{m}\psi_{m}(z)e^{-2L\hat{\kappa}_m} + \sum_{m=0}^{\infty}c^s_{m}\psi_{m}(z)
=\sum_{m=0}^{\infty}d^s_{m} \phi_{m}\left( z\right)
</math>
</center>
<center>
<math>
-\sum_{m=0}^{\infty}b^s_{m}\hat{\kappa}_{m}\psi
_{m}(z)e^{-2L\hat{\kappa}_m} + \sum_{m=0}^{\infty}c^s_{m}\hat{\kappa}_{m}\psi
_{m}(z)
= \sum_{m=0}^{\infty}d^s_{m} \tanh(\hat{k}_m L_1) \hat{k}_m\phi_{m}\left( z\right)
</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>
and we always take the positive real root or the root with positive imaginary part.

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^s_{l}A_{l}
=\sum_{m=0}^{\infty}b^s_{m}B_{ml}
+ \sum_{m=0}^{\infty}c^s_{m}B_{ml}e^{-2L\hat{\kappa}_m}
</math>
</center>
<center>
<math>
-\hat{k}_{0}A_{0}\delta_{0l}+a^s_{l}\hat{k}_{l}A_l
= - \sum_{m=0}^{\infty}b^s_{m}\hat{\kappa}_{m}B_{ml}
+ \sum_{m=0}^{\infty}c^s_{m}\hat{\kappa}_{m}B_{ml} e^{-2L\hat{\kappa}}
</math>
</center>
<center>
<math>
\sum_{m=0}^{\infty}b^s_{m}B_{ml}e^{-2L\hat{\kappa}_m}
+ \sum_{m=0}^{\infty}c^s_{m}B_{ml}
=d^s_l A_l
</math>
</center>
<center>
<math>
- \sum_{m=0}^{\infty}b^s_{m}\hat{\kappa}_{m}B_{ml} e^{-2L\hat{\kappa}_m}
+ \sum_{m=0}^{\infty}c^s_{m}\hat{\kappa}_{m}B_{ml}
= d^s_l \tanh(\hat{k}_l L_1) \hat{k}_l A_l
</math>
</center>
and these are solved exactly as before. The solution for the anti-symmetric
potential is found in a similar fashion.

= 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/two_finite_docks_symmetry.m two_finite_docks_symmetry.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]]

Eigenfunction Matching for a Finite Dock

2008-06-25T18:14:50Z

F. Bonnefoy: /* Governing Equations */

= 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{\cos\kappa_{m} x}{\cos \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}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</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}\tan\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.

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_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</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}\cot\kappa_{m}L B_{ml}
</math>
</center>
(for full details of this derivation see [[Eigenfunction Matching for a Semi-Infinite Dock]])

Eigenfunction Matching for a Finite Dock

2008-06-25T18:14:22Z

F. Bonnefoy: /* Governing Equations */

= 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{\cos\kappa_{m} x}{\cos \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}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</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}\tan\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.

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_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</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}\cot\kappa_{m}L B_{ml}
</math>
</center>
(for full details of this derivation see [[Eigenfunction Matching for a Semi-Infinite Dock]])

Eigenfunction Matching for a Finite Dock

2008-06-25T18:07:24Z

F. Bonnefoy: /* Introduction */

= 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 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>
\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{\cos\kappa_{m} x}{\cos \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}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</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}\tan\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.

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_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}
</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}\cot\kappa_{m}L B_{ml}
</math>
</center>
(for full details of this derivation see [[Eigenfunction Matching for a Semi-Infinite Dock]])

File:Antisymmetric.jpg

2008-06-25T13:53:02Z

F. Bonnefoy:

File:Symmetric FB.jpg

2008-06-25T13:33:52Z

F. Bonnefoy:

Eigenfunction Matching for a Finite Dock

2008-06-25T13:18:10Z

F. Bonnefoy: /* Introduction */

File:Finite dock.jpg

2008-06-25T13:17:11Z

F. Bonnefoy:

Eigenfunction Matching for a Semi-Infinite Dock

2008-06-25T13:04:34Z

F. Bonnefoy: /* Introduction */

= 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|semi-infinite submerged 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>
\phi_{z}=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==

We now separate variables and write the potential as
<center>
<math>
\phi(x,z)=\zeta(z)\rho(x)\,
</math>
</center>
Applying Laplace's equation we obtain
<center>
<math>
\zeta_{zz}+\mu^{2}\zeta=0.\,
</math>
</center>
We then use the boundary condition at <math>z=-h</math>, which is
the same for all <math>x</math> to write
<center>
<math>
\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
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==

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==

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>n</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_{n=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 mutiple equation (3) by <math>k_l</math> and subtract equation (4)
we obtain
<center>
<math>
2k_{0}A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml}
</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 (10) we set the upper limit of <math>l</math> to
be <math>M</math>. In terms of matrix, we obtain
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l\delta_{0l}&\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>. 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).

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)=\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
[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/semiinfinite_dock.m semiinfinite_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]]

Eigenfunction Matching for a Semi-Infinite Dock

2008-06-25T13:03:53Z

F. Bonnefoy: /* Separation of variables */

= 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|semi-infinite submerged dock]]

[[Image:semiinfinite_dock.jpg|thumb|center|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>
\phi_{z}=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==

We now separate variables and write the potential as
<center>
<math>
\phi(x,z)=\zeta(z)\rho(x)\,
</math>
</center>
Applying Laplace's equation we obtain
<center>
<math>
\zeta_{zz}+\mu^{2}\zeta=0.\,
</math>
</center>
We then use the boundary condition at <math>z=-h</math>, which is
the same for all <math>x</math> to write
<center>
<math>
\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
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==

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==

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>n</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_{n=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 mutiple equation (3) by <math>k_l</math> and subtract equation (4)
we obtain
<center>
<math>
2k_{0}A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml}
</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 (10) we set the upper limit of <math>l</math> to
be <math>M</math>. In terms of matrix, we obtain
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l\delta_{0l}&\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>. 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).

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)=\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
[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/semiinfinite_dock.m semiinfinite_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]]

Eigenfunction Matching for a Semi-Infinite Dock

2008-06-25T13:02:13Z

F. Bonnefoy: /* Introduction */

= 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|semi-infinite submerged dock]]

[[Image:semiinfinite_dock.jpg|thumb|center|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>
\phi_{z}=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==

We now separate variables and write the potential as
<center>
<math>
\phi(x,z)=\zeta(z)\rho(x)
</math>
</center>
Applying Laplace's equation we obtain
<center>
<math>
\zeta_{zz}+\mu^{2}\zeta=0.
</math>
</center>
We then use the boundary condition at <math>z=-h</math>, which is
the same for all <math>x</math> to write
<center>
<math>
\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
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==

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==

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>n</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_{n=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 mutiple equation (3) by <math>k_l</math> and subtract equation (4)
we obtain
<center>
<math>
2k_{0}A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml}
</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 (10) we set the upper limit of <math>l</math> to
be <math>M</math>. In terms of matrix, we obtain
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l\delta_{0l}&\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>. 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).

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)=\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
[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/semiinfinite_dock.m semiinfinite_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]]

Eigenfunction Matching for a Semi-Infinite Dock

2008-06-25T13:00:34Z

F. Bonnefoy: /* Introduction */

= 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|semi-infinite submerged dock]]

[[Image:semiinfinite_dock.jpg|thumb|right|600px|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>
\phi_{z}=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==

We now separate variables and write the potential as
<center>
<math>
\phi(x,z)=\zeta(z)\rho(x)
</math>
</center>
Applying Laplace's equation we obtain
<center>
<math>
\zeta_{zz}+\mu^{2}\zeta=0.
</math>
</center>
We then use the boundary condition at <math>z=-h</math>, which is
the same for all <math>x</math> to write
<center>
<math>
\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
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==

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==

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>n</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_{n=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 mutiple equation (3) by <math>k_l</math> and subtract equation (4)
we obtain
<center>
<math>
2k_{0}A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml}
</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 (10) we set the upper limit of <math>l</math> to
be <math>M</math>. In terms of matrix, we obtain
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l\delta_{0l}&\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>. 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).

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)=\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
[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/semiinfinite_dock.m semiinfinite_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]]

File:Semiinfinite dock.jpg

2008-06-25T12:48:37Z

F. Bonnefoy:

Eigenfunction Matching for a Semi-Infinite Dock

2008-06-25T09:30:32Z

F. Bonnefoy: /* Numerical Solution */

= 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|semi-infinite submerged 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>
\phi_{z}=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==

We now separate variables and write the potential as
<center>
<math>
\phi(x,z)=\zeta(z)\rho(x)
</math>
</center>
Applying Laplace's equation we obtain
<center>
<math>
\zeta_{zz}+\mu^{2}\zeta=0.
</math>
</center>
We then use the boundary condition at <math>z=-h</math>, which is
the same for all <math>x</math> to write
<center>
<math>
\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
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==

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==

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>n</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_{n=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 mutiple equation (3) by <math>k_l</math> and subtract equation (4)
we obtain
<center>
<math>
2k_{0}A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml}
</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 (10) we set the upper limit of <math>l</math> to
be <math>M</math>. In terms of matrix, we obtain
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l\delta_{0l}&\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>. 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).

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)=\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
[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/semiinfinite_dock.m semiinfinite_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]]

Linear and Second-Order Wave Theory

2008-06-24T07:27:07Z

F. Bonnefoy: /* Derivation using Bernoulli's equation */

We saw in [[Conservation Laws And Boundary Conditions]] that the potential flow model for wave progation is given Laplaces equation plus the free-surface condtions. In this section we present the linear and second order theory for these equations. The linear theory is valid for small wave heights and the second order theory is an improvement on this. However, neither of these theories work for very steep waves and of course the potential theory breaks down once the wave begins to break and completely different methods are required in this situation.

= Linerization of Free-surface Conditions =

We use [http://en.wikipedia.org/wiki/Perturbation_theory perturbation theory] to expand the solution as follows
<center><math>\zeta = \zeta_1 + \zeta_2 + \zeta_3 + \cdots </math></center>
<center><math> \Phi = \Phi_1 + \Phi_2 + \Phi_3 + \cdots </math></center>
where we are assuming that there exists a small parameter (the wave slope) and that with respect to this the
<math>\Phi_i</math> is proportional to <math>\epsilon^i</math> or <math>O(\epsilon^i)</math>. We then derive the boundary value problem for <math> \zeta_i,\Phi_i </math>. Rarely we need to go beyond <math> i = 3 </math> (in fact it is unlikely that the terms beyond
this will improve the accuracy.

In this section we will only derive the free-surface conditions up to second order. Remember that <math>\nabla^2 \Phi_i =0</math> for all <math>i</math>
We expand the kinematic and dynamic free surface conditions about the <math>Z=0</math> plane and derive statements for the unknown pairs <math> (\Phi_1,\zeta_1 </math> and <math> (\Phi_2, \zeta_2) </math> at <math> Z=0 </math>. The same technique can be used to linearize the body boundary condition at <math> U=0 </math> (zero speed) and <math> U>0 </math> (forward speed).

= Kinematic condition =

The fully non-linear kinematic condition was derived in [[Conservation Laws and Boundary Conditions]] and we begin with this equation
<center><math> \left ( \frac{\partial \zeta}{\partial t} + \nabla \Phi \cdot \nabla \zeta \right )_{Z=\zeta} = \left ( \frac{\partial \Phi}{\partial Z} \right )_{Z=\zeta} </math></center>
We expand this equation about <math>\zeta = 0</math>, which we can do because we have assumed that the slope is small. In fact the slope is our parameter <math>\epsilon</math>. It is obvious at this point that the theory does not apply to very steep waves. This gives us the following equation
<center><math> \left( \frac{\partial\zeta}{\partial t} + \nabla \Phi \cdot \nabla \zeta \right)_{Z=0} + \zeta \frac{\partial}{\partial Z} \left( \frac{\partial\zeta}{\partial t} + \nabla \Phi \cdot \nabla \zeta \right)_{Z=0} + \cdots </math></center><br>
<center><math> = \left( \frac{\partial\Phi}{\partial Z} \right)_{Z=0} + \zeta \left( \frac{\partial^2 \Phi}{\partial Z^2} \right)_{Z=0} + \cdots </math></center>
where we have only taken the first order expansion. We then substitute our expressions
<center> <math>
\begin{matrix}
\zeta = \zeta_1 + \zeta_2 + \cdots \\
\Phi = \Phi_1 + \Phi_2 + \cdots
\end{matrix}
</math></center>
and keep terms of <math>\ O(\varepsilon), \ O(\varepsilon^2)</math>, remembering that <math>\zeta_1\Phi_1</math> is <math>O(\varepsilon^2)</math> etc.

= Dynamic condition =

The fully non-linear Dynamic condition was derived in [[Conservation Laws and Boundary Conditions]] and is given by
<center><math>
\zeta (X,Y,t) = -\frac{1}{g} \left( \frac{\partial\Phi}{\partial t} + \frac{1}{2} \nabla\Phi \cdot \nabla\Phi \right)_{Z=\zeta}
</math></center>

<center><math>
\left . \begin{matrix}
\zeta = \frac{1}{g} \left( \frac{\partial\Phi}{\partial t} + \frac{1}{2} \nabla\Phi \cdot \nabla\Phi \right)_{Z=0}\\
\frac{1}{g} \zeta \frac{\partial}{\partial Z} \left( \frac{\partial\Phi}{\partial t} + \frac{1}{2} \nabla\Phi \cdot \nabla\Phi \right)_{Z=0} + \cdots
\end{matrix} \right \}
\begin{matrix}
\zeta = \zeta_1 +\zeta_2 + \cdots \\
\Phi = \Phi_1 + \Phi_2 + \cdots
\end{matrix}
</math></center>

= Linear problem =

The linear problem is the <math>O(\varepsilon)</math> problem derived by equating the terms which are proportional to <math>\epsilon</math>.

This can be done straight forwardly and gives the following expressions

<center><math> \frac{\partial\zeta_1}{\partial t} = \frac{\partial\Phi_1}{\partial Z} , \ Z=0; </math></center>
which follows from the Kinematic equation and
<center><math> \zeta_1 = -\frac{1}{g} \frac{\partial\Phi_1}{\partial t}, \ Z=0; </math></center>
which follows from the Dynamic equation. These are the linear free surface conditions.

== Derivation using Bernoulli's equation ==

The pressure from Bernoulli, <math> \omega </math> constant terms set equal to zero, at a fixed point in the fluid domain at <math> \vec{X}=(X,Y,Z) </math> is given by:
<center><math> P = - \rho \left( \frac{\partial\Phi}{\partial t} + \frac{1}{2} \nabla\Phi \cdots \nabla\Phi + gZ \right); </math></center>
When then make the perturbation expansion for the potential and the pressure
<center><math> \Phi = \Phi_1 + \Phi_2 + \cdots </math></center>
and
<center><math> P = P_0 + P_1 + P_2 + \cdots </math></center>.

This allows us to derive
<center><math> P_0 = -\rho g Z </math></center>
which is called the Hydrostatic pressure and
<center><math> P_1 = - \rho \frac{\partial\Phi_1}{\partial t} </math></center>
which is the linear pressure.

== Classical linear free surface condition ==
If we eliminate <math> \zeta_1 </math> from the kinematic and dynamic free surface conditions, we obtain the classical linear free surface condition:
<center><math> \begin{cases}
\frac{\partial^2\Phi_1}{\partial t^2} + g \frac{\partial\Phi_1}{\partial Z} = 0, \qquad Z=0\\
\zeta_1 = - \frac{1}{g} \frac{\partial\Phi_1}{\partial t}, \qquad Z=0
\end{cases} </math></center>

With:

<center><math> P_1 = - \rho \frac{\partial\Phi_1}{\partial t}, \qquad \mbox{At some fixed point} \ \vec X </math></center>

Note that on <math> Z=0, \ P_1 \ne 0 </math> in fact it can obtained from the expressions above in the form

<center><math> P_1 = -\rho g \zeta_1, \qquad Z=0 </math></center>

So linear theory states that the linear perturbation pressure on the <math> Z=0 \, </math> plane due to a surface wave disturbance is equal to the positive (negative) "hydrostatic" pressure induced by the positive (negative) wave elevation <math> \zeta_1 \, </math>.

= Second-order problem =

The second order equations can also be derived straight forwardly. The kinematic condition is
<center><math> \frac{\partial\zeta_2}{\partial t} + \nabla\Phi_1 \cdot \nabla\zeta_1 = \frac{\partial\Phi_2}{\partial Z} + \zeta_1 \frac{\partial^2 \Phi_1}{\partial Z^2}, \quad Z=0 </math></center>
and the dynamic condition
<center><math> \zeta_2 = - \frac{1}{g} \left( \frac{\partial\Phi_2}{\partial t} + \frac{1}{2} \nabla\Phi_1 \cdot \nabla\Phi_1 \right)_{Z=0} - \frac{1}{g} \zeta_1 \frac{\partial^2\Phi_1}{\partial Z \partial t}, \quad Z=0 </math></center>

Alternatively, the known linear terms may be moved in the right-hand side as forcing functions, leading to:

== Kinematic second-order condition ==
<center>
<math> \frac{\partial\zeta_2}{\partial t} - \frac{\partial\Phi_2}{\partial Z} = \zeta_1 \frac{\partial^2 \Phi_1}{\partial Z^2} - \nabla\Phi_1 \cdot \nabla\zeta_1; \quad Z=0 </math>
</center>

== Dynamic second-order condition ==

<center>
<math> \zeta_2 + \frac{1}{g} \frac{\partial\Phi_2}{\partial t} = - \frac{1}{g} \left( \frac{1}{2} \nabla\Phi_1 \cdot \nabla\Phi_1 + \zeta_1 \frac{\partial^2\Phi_1}{\partial Z \partial t} \right)_{Z=0} </math>
</center>
where the second order pressure is given by
<center>
<math> P_2 = -\rho \left( \frac{\partial\Phi_2}{\partial t} + \frac{1}{2} \nabla\Phi_1 \cdot \nabla\Phi_1 \right); \quad \mbox{at} \ \vec X. </math>
</center>

The very attractive feature of second order surface wave theory is that it allows the prior solution of the linear problem which is often possible analytically and numerically.
The linear solution is then used as a forcing function for the solution of the second order problem. This is often possible analytically and in most cases numerically in the absence or presence of bodies.
Linear and second-order theories are also very appropriate to use for the modeling of surface waves as stochastic processes.
Both theories are very useful in practice, particularly in connection with wave-body interactions.

-----

This article is based on the MIT open course notes and the original article can be found
[http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-24Spring-2002/0B7683D3-9B31-453E-B98F-9F71A3C36C58/0/lecture2.pdf here]

[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]

[[Category:Linear Water-Wave Theory]]
[[Category:Non-linear Water-Wave Theory]]

Eigenfunction Matching for a Semi-Infinite Dock

2008-06-24T07:25:54Z

F. Bonnefoy: /* Numerical Solution */

= 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|semi-infinite submerged 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>
\phi_{z}=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==

We now separate variables and write the potential as
<center>
<math>
\phi(x,z)=\zeta(z)\rho(x)
</math>
</center>
Applying Laplace's equation we obtain
<center>
<math>
\zeta_{zz}+\mu^{2}\zeta=0.
</math>
</center>
We then use the boundary condition at <math>z=-h</math>, which is
the same for all <math>x</math> to write
<center>
<math>
\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
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==

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==

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>n</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_{n=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 mutiple equation (3) by <math>k_l</math> and subtract equation (4)
we obtain
<center>
<math>
2k_{0}A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml}
</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 (10) we set the upper limit of <math>l</math> to
be <math>M</math>. In terms of matrix, we obtain
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l\delta_{0l}&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{nl}&\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_{n}) \, B_{nl}&\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>. 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).

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)=\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
[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/semiinfinite_dock.m semiinfinite_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]]

Eigenfunction Matching for a Semi-Infinite Dock

2008-06-24T07:25:21Z

F. Bonnefoy: /* Numerical Solution */

= 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|semi-infinite submerged 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>
\phi_{z}=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==

We now separate variables and write the potential as
<center>
<math>
\phi(x,z)=\zeta(z)\rho(x)
</math>
</center>
Applying Laplace's equation we obtain
<center>
<math>
\zeta_{zz}+\mu^{2}\zeta=0.
</math>
</center>
We then use the boundary condition at <math>z=-h</math>, which is
the same for all <math>x</math> to write
<center>
<math>
\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
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==

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==

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>n</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_{n=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 mutiple equation (3) by <math>k_l</math> and subtract equation (4)
we obtain
<center>
<math>
2k_{0}A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml}
</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 (10) we set the upper limit of <math>l</math> to
be <math>M</math>. In terms of matrix, we obtain
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0 \quad \cdots&0\\
0&&\\
\vdots&A_l\delta_{0l}&\vdots\\
&&0\\
0&\cdots \quad 0 &A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&\cdots&-B_{0M}\\
&&\\
\vdots&-B_{nl}&\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_{n}) \, B_{nl}&\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>. 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).

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)=\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
[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/semiinfinite_dock.m semiinfinite_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]]

Eigenfunction Matching for a Semi-Infinite Dock

2008-06-24T07:14:58Z

F. Bonnefoy: /* Numerical Solution */

= 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|semi-infinite submerged 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>
\phi_{z}=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==

We now separate variables and write the potential as
<center>
<math>
\phi(x,z)=\zeta(z)\rho(x)
</math>
</center>
Applying Laplace's equation we obtain
<center>
<math>
\zeta_{zz}+\mu^{2}\zeta=0.
</math>
</center>
We then use the boundary condition at <math>z=-h</math>, which is
the same for all <math>x</math> to write
<center>
<math>
\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
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==

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==

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>n</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_{n=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 mutiple equation (3) by <math>k_l</math> and subtract equation (4)
we obtain
<center>
<math>
2k_{0}A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml}
</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 (10) we set the upper limit of <math>l</math> to
be <math>M</math>. In terms of matrix, we obtain
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
A_0&0&\cdots&&0\\
0&\ddots&\ddots&&\vdots\\
\vdots&\ddots&A_l\delta_{0l}&&\\
&&&\ddots&0\\
0&\cdots&&0&A_M
\end{bmatrix}
&
\begin{bmatrix}
-B_{00}&-B_{01}&\cdots&&-B_{0M}\\
-B_{10}&\ddots&&&\vdots\\
\vdots&&-B_{nl}&&\\
&&&\ddots&\\
-B_{M0}&\cdots&&&-B_{MM}
\end{bmatrix}
\\
\begin{bmatrix}
0&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots&0
\end{bmatrix}
&
\begin{bmatrix}
\ddots&&0\\
&(k_l + \kappa_{n}) \, B_{nl}&\\
0&&\ddots
\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>. 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).

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)=\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
[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/semiinfinite_dock.m semiinfinite_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]]

Eigenfunction Matching for a Semi-Infinite Dock

2008-06-24T07:03:29Z

F. Bonnefoy: /* Numerical Solution */

= 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|semi-infinite submerged 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>
\phi_{z}=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==

We now separate variables and write the potential as
<center>
<math>
\phi(x,z)=\zeta(z)\rho(x)
</math>
</center>
Applying Laplace's equation we obtain
<center>
<math>
\zeta_{zz}+\mu^{2}\zeta=0.
</math>
</center>
We then use the boundary condition at <math>z=-h</math>, which is
the same for all <math>x</math> to write
<center>
<math>
\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
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==

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==

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>n</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_{n=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 mutiple equation (3) by <math>k_l</math> and subtract equation (4)
we obtain
<center>
<math>
2k_{0}A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml}
</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 (10) we set the upper limit of <math>l</math> to
be <math>M</math>. In terms of matrix, we obtain
<center>
<math>
\begin{bmatrix}
\begin{bmatrix}
&&\\
&A_l\delta_{0l}&\\
&&
\end{bmatrix}
&
\begin{bmatrix}
&&\\
&-B_{nl}&\\
&&
\end{bmatrix}
\\
\begin{bmatrix}
&&\\
&0&\\
&&
\end{bmatrix}
&
\begin{bmatrix}
&&\\
&(k_l + \kappa_{n}) \, B_{nl}&\\
&&
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
a_{0} \\
a_{1} \\
\vdots \\
a_N \\
b_{0}\\
b_1 \\
\vdots \\
b_N
\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>. 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).

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)=\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
[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/semiinfinite_dock.m semiinfinite_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]]

Eigenfunction Matching for a Semi-Infinite Dock

2008-06-24T06:54:22Z

F. Bonnefoy: /* Numerical Solution */

= 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|semi-infinite submerged 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>
\phi_{z}=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==

We now separate variables and write the potential as
<center>
<math>
\phi(x,z)=\zeta(z)\rho(x)
</math>
</center>
Applying Laplace's equation we obtain
<center>
<math>
\zeta_{zz}+\mu^{2}\zeta=0.
</math>
</center>
We then use the boundary condition at <math>z=-h</math>, which is
the same for all <math>x</math> to write
<center>
<math>
\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
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==

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==

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>n</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_{n=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 mutiple equation (3) by <math>k_l</math> and subtract equation (4)
we obtain
<center>
<math>
2k_{0}A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml}
</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 (10) 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.
<center>
<math>
\left[
\begin{array}{ccccccc}
\ulcorner &&\urcorner&&\ulcorner &&\urcorner\\
&A_l\delta_{0l}&&&&-B_{nl}&\\
\llcorner &&\lrcorner&&\llcorner &&\lrcorner\\
\\
\ulcorner &&\urcorner&&\ulcorner &&\urcorner\\
&0&&&&(k_l + \kappa_{n}) \, B_{nl}&\\
\llcorner &&\lrcorner&&\llcorner &&\lrcorner\\
\end{array}
\right]
\left[
\begin{array}{c}
a_{0} \\
a_{1} \\
\vdots \\
a_N \\
b_{0}\\
b_1 \\
\vdots \\
b_N
\end{array}
\right] \egal
\left[
\begin{array}{c}
- A_{0} \\
0 \\
\vdots \\
0 \\
2k_{0}A_{0} \\
0 \\
\vdots \\
0
\end{array}
\right]
</center>
</math>

= 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).

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)=\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
[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/semiinfinite_dock.m semiinfinite_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]]

Eigenfunction Matching for a Finite Dock

2008-04-15T07:47:28Z

F. Bonnefoy: /* Solution with Waves Incident at an Angle */

Eigenfunction Matching for a Finite Dock

2008-04-15T07:46:50Z

F. Bonnefoy: /* An infinite dimensional system of equations */

Eigenfunction Matching for a Finite Dock

2008-04-12T07:34:10Z

F. Bonnefoy: /* An infinite dimensional system of equations */

Talk:Eigenfunction Matching for a Finite Dock

2008-04-12T07:14:44Z

F. Bonnefoy:

== Broken link ==

The link for the finite.m file is broken. Or maybe the file doesn't exist yet.

Talk:Eigenfunction Matching for a Finite Dock

2008-04-11T16:01:16Z

F. Bonnefoy:

== Broken link ==

The link for the finite.m file is broken. Or maybe the file doesn't exist yet.

== Expression of the potential under the dock ==

It seems that the eigenfunctions
<math>
\frac{L\pm x}{2L}\psi_{0}(z)
</math>
do not satisfy the Laplace equation. Nor do any linear combination of them so it seems to me that we have to write <math>b_0=c_0=0</math> which means removing them from the potential expression.

Eigenfunctions <math>\frac{L\pm x}{2L}</math> work better but they will cause problems for infinite depth...

Talk:Eigenfunction Matching for a Finite Dock

2008-04-11T15:59:18Z

F. Bonnefoy: Expression of the potential under the dock

Talk:Eigenfunction Matching for a Finite Dock

2008-04-11T15:55:53Z

F. Bonnefoy:

== Broken link ==

The link for the finite.m file is broken. Or maybe the file doesn't exist yet.

Talk:Eigenfunction Matching for a Finite Dock

2008-04-11T15:55:06Z

F. Bonnefoy:

The link for the finite.m file is broken. Or maybe the file doesn't exist yet.

test

Talk:Eigenfunction Matching for a Finite Dock

2008-04-11T15:22:14Z

F. Bonnefoy:

The link for the finite.m file is broken. Or maybe the file doesn't exist yet.

Eigenfunction Matching for a Finite Dock

2008-04-11T15:21:23Z

F. Bonnefoy: /* Matlab Code */

Eigenfunction Matching for a Finite Dock

2008-04-11T15:21:05Z

F. Bonnefoy: /* Matlab Code */

Eigenfunction Matching for a Semi-Infinite Dock

2008-04-11T15:20:19Z

F. Bonnefoy: /* Matlab Code */

= 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.

=Governing Equations=

We begin with the [[Frequency Domain Problem]] for a dock which occupies
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<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==

We now separate variables and write the potential as
<center>
<math>
\phi(x,z)=\zeta(z)\rho(x)
</math>
</center>
Applying Laplace's equation we obtain
<center>
<math>
\zeta_{zz}+\mu^{2}\zeta=0.
</math>
</center>
We then use the boundary condition at <math>z=-h</math>, which is
the same for all <math>x</math> to write
<center>
<math>
\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
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 plate covered region respectively.

==Incident potential==

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==

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>n</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_{n=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 mutiple equation (3) by <math>k_l</math> and subtract equation (4)
we obtain
<center>
<math>
2k_{0}A_{0}\delta_{0l}
=\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml}
</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 (10) 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).

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)=\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
[http://www.math.auckland.ac.nz/~meylan/code/semiinfinite_dock.m semiinfinite_dock.m]

== Additional code ==

This program requires
[http://www.math.auckland.ac.nz/~meylan/code/dispersion_free_surface.m dispersion_free_surface.m]
to run

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

User:F. Bonnefoy

2008-04-06T22:48:52Z

F. Bonnefoy:

#REDIRECT [[ F. Bonnefoy]]