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

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{{complete pages}}
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= Introduction =
 
= Introduction =
  
This is the finite length version of the [[Eigenfunction Matching for a Submerged Dock]]. The full
+
This is the finite length version of the [[Eigenfunction Matching for a Submerged Semi-Infinite Dock]]. The full
 
theory is not presented here, and details of the matching method can be found in  
 
theory is not presented here, and details of the matching method can be found in  
[[Eigenfunction Matching for a Submerged Dock]] and [[Eigenfunction Matching for a Finite Dock]]
+
[[Eigenfunction Matching for a Submerged Semi-Infinite Dock]] and [[Eigenfunction Matching for a Finite Dock]]
  
 
=Governing Equations=
 
=Governing Equations=
Line 20: Line 23:
 
<center>
 
<center>
 
<math>
 
<math>
\phi_{z}=0, \,\, z=-h,
+
\partial_{z} \phi=0, \,\, z=-h,
 
</math>
 
</math>
 
</center>
 
</center>
Line 28: Line 31:
 
<center>
 
<center>
 
<math>
 
<math>
\partial_z\phi=0, \,\, z=-d,\,x>0,
+
\partial_z\phi=0, \,\, z=-d,\,-L<x<L,
 
</math>
 
</math>
 
</center>
 
</center>
Line 39: Line 42:
 
=Solution Method=
 
=Solution Method=
  
We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the three regions, <math>x<0</math>
+
We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the four regions, {<math>x<-L \,</math>}, {<math>x>L \,</math>}, {<math>-d<z<0,\,\,-L<x<L</math>}, and {<math>-h<z<-d,\,\,-L<x<L</math>}. The first three regions use the free-surface eigenfunction
<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 last uses dock eigenfunctions. Details can be found in [[Eigenfunction Matching for a Semi-Infinite Dock]].
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>
 
The incident potential is a wave of amplitude <math>A</math>
Line 48: Line 50:
 
<center>
 
<center>
 
<math>
 
<math>
\phi^{\mathrm{I}}  =e^{-k_{0}x}\phi_{0}\left(
+
\phi^{\mathrm{I}}  =e^{-k_{0}^{h}(x+L)}\phi_{0}\left(
 
z\right)  
 
z\right)  
 
</math>
 
</math>
Line 57: Line 59:
 
<center>
 
<center>
 
<math>
 
<math>
\phi(x,z)=e^{-k_{0}^h x}\phi_{0}^h\left(
+
\phi(x,z)=e^{-k_{0}^h (x+L)}\phi_{0}^h\left(
z\right) + \sum_{m=0}^{\infty}a_{m}e^{k_{m}^h x}\phi_{m}^h(z), \;\;x<0
+
z\right) + \sum_{m=0}^{\infty}a_{m}e^{k_{m}^h (x+L)}\phi_{m}^h(z), \;\;x<-L
 
</math>
 
</math>
 
</center>
 
</center>
Line 64: Line 66:
 
<math>
 
<math>
 
\phi(x,z)= \sum_{m=0}^{\infty}b_{m}
 
\phi(x,z)= \sum_{m=0}^{\infty}b_{m}
e^{-k_{m}^d (x)}\phi_{m}^d(z)
+
e^{-k_{m}^d (x+L)}\phi_{m}^d(z)
, \;\;-d<z<0,\,\,x>0
+
+ \sum_{m=0}^{\infty}c_{m}
 +
e^{k_{m}^d (x-L)}\phi_{m}^d(z)
 +
, \;\;-d<z<0,\,\,-L<x<L
 
</math>
 
</math>
 
</center>
 
</center>
Line 71: Line 75:
 
<center>
 
<center>
 
<math>
 
<math>
\phi(x,z)= \sum_{m=0}^{\infty}c_{m}
+
\phi(x,z)= d_0 \frac{L-x}{2 L} + \sum_{m=1}^{\infty}d_{m}
e^{\kappa_{m} x}\psi_{m}(z)
+
e^{-\kappa_{m} (x+L)}\psi_{m}(z)
, \;\;-h<z<-d,\,\,x>0
+
+ e_0 \frac{x+L}{2 L} +  
</math>
+
\sum_{m=1}^{\infty}e_{m}
</center>
+
e^{\kappa_{m} (x-L)}\psi_{m}(z)
where <math>a_{m}</math> and <math>b_{m}</math>
+
, \;\;-h<z<-d,\,\,-L<x<L
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>
 
</math>
 
</center>
 
</center>
 
<center>
 
<center>
 
<math>
 
<math>
-k_0^h\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty}
+
\phi(x,z)= \sum_{m=0}^{\infty}f_{m}e^{-k_{m}^h (x-L)}\phi_{m}^h(z), \;\;L<x
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>
 
</math>
 
</center>
 
</center>
  
=Numerical Solution=
+
The definition of all terms can be found in [[Eigenfunction Matching for Submerged Semi-Infinite Dock]],
 
+
as can the solution method and the method to extend the solution to waves incident at an angle.
The standard method to solve these equations (from [[Linton and Evans 1991]]) 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_n =
 
\begin{cases}
 
\psi_{n}(z),\,\,\,-h<z<-d \\
 
0,\,\,\,-d<z< 0
 
\end{cases}
 
</math>
 
</center>
 
for <math>0 \leq n \leq M - 1 </math>
 
<center>
 
<math>
 
\chi_{n+M} =
 
\begin{cases}
 
0,\,\,\,-h<z<-d \\
 
\phi_{n}^{d}(z),\,\,\,-d<z< 0
 
\end{cases}
 
</math>
 
</center>
 
for <math>0 \leq n \leq N-M </math>
 
and we choose the values of <math>N</math> so that we have the <math>N+1</math> smallest values
 
of <math>k_n</math> and <math>\kappa_n</math> (with the proviso that we have at least one from each).
 
 
 
We truncate the equations and write
 
 
 
<center>
 
<math>
 
\phi_{0}^h\left(  z\right) + \sum_{m=0}^{N}
 
a_{m} \phi_{m}^h\left(  z\right)
 
=\sum_{m=0}^{N}b_{m} \chi_m,
 
</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}^{N}k^{\prime}_m b_{m}\chi_{m}
 
</math>
 
</center>
 
where <math>k^{\prime}_m</math> is either <math>k^{d}_q</math> or  <math>\kappa_q</math>
 
as appropriate.
 
 
 
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_{0}\delta_{0q} + a_{q}A_{q}
 
= \sum_{m=0}^{N} b_m B^{\prime}_{mq}
 
</math>
 
</center>
 
<center>
 
<math>
 
-k_{0}^h A_{0}\delta_{0q} + k_{q}^h a_{q}A_{q}
 
= \sum_{m=0}^{N} k^{\prime}_m b_m B^{\prime}_{mq}
 
</math>
 
</center>
 
where <math>B^{\prime}_{mq}</math> is made from <math>B_{mq}</math> or <math>C_{mq}</math> as appropriate.
 
 
 
= 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 =
 
= Matlab Code =
  
 
A program to calculate the coefficients for the submerged semi-infinite dock problems can be found here
 
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]
+
[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/submerged_finite_dock.m submerged_finite_dock.m]
  
 
== Additional code ==
 
== Additional code ==
  
 
This program requires
 
This program requires
[http://www.math.auckland.ac.nz/~meylan/code/dispersion/dispersion_free_surface.m dispersion_free_surface.m]
+
* {{free surface dispersion equation code}}
to run
+
 
  
 
[[Category:Eigenfunction Matching Method]]
 
[[Category:Eigenfunction Matching Method]]
 
[[Category:Pages with Matlab Code]]
 
[[Category:Pages with Matlab Code]]
 
[[Category:Complete Pages]]
 
[[Category:Complete Pages]]

Latest revision as of 05:54, 1 September 2009



Introduction

This is the finite length version of the Eigenfunction Matching for a Submerged Semi-Infinite Dock. The full theory is not presented here, and details of the matching method can be found in Eigenfunction Matching for a Submerged Semi-Infinite Dock and Eigenfunction Matching for a Finite Dock

Governing Equations

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

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

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

[math]\displaystyle{ \partial_z\phi=\alpha\phi, \,\, z=0, }[/math]

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

We must also apply the Sommerfeld Radiation Condition as [math]\displaystyle{ |x|\rightarrow\infty }[/math]. This essentially implies that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave and a wave propagating away.

Solution Method

We use separation of variables in the four regions, {[math]\displaystyle{ x\lt -L \, }[/math]}, {[math]\displaystyle{ x\gt L \, }[/math]}, {[math]\displaystyle{ -d\lt z\lt 0,\,\,-L\lt x\lt L }[/math]}, and {[math]\displaystyle{ -h\lt z\lt -d,\,\,-L\lt x\lt L }[/math]}. The first three regions use the free-surface eigenfunction and the last uses dock eigenfunctions. Details can be found in Eigenfunction Matching for a Semi-Infinite Dock.

The incident potential is a wave of amplitude [math]\displaystyle{ A }[/math] in displacement travelling in the positive [math]\displaystyle{ x }[/math]-direction. The incident potential can therefore be written as

[math]\displaystyle{ \phi^{\mathrm{I}} =e^{-k_{0}^{h}(x+L)}\phi_{0}\left( z\right) }[/math]

The potential can be expanded as

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

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

and

[math]\displaystyle{ \phi(x,z)= d_0 \frac{L-x}{2 L} + \sum_{m=1}^{\infty}d_{m} e^{-\kappa_{m} (x+L)}\psi_{m}(z) + e_0 \frac{x+L}{2 L} + \sum_{m=1}^{\infty}e_{m} e^{\kappa_{m} (x-L)}\psi_{m}(z) , \;\;-h\lt z\lt -d,\,\,-L\lt x\lt L }[/math]

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

The definition of all terms can be found in Eigenfunction Matching for Submerged Semi-Infinite Dock, as can the solution method and the method to extend the solution to waves incident at an angle.

Matlab Code

A program to calculate the coefficients for the submerged semi-infinite dock problems can be found here submerged_finite_dock.m

Additional code

This program requires