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

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{{complete pages}}
 +
 +
 
= Introduction =
 
= Introduction =
  
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<center>
 
<center>
 
<math>
 
<math>
\phi_{z}=0, \,\, z=-h,
+
\partial_{z} \phi=0, \,\, z=-h,
 
</math>
 
</math>
 
</center>
 
</center>
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<center>
 
<center>
 
<math>
 
<math>
\partial_z\phi=0, \,\, z=-d,\,-L<x>L,
+
\partial_z\phi=0, \,\, z=-d,\,-L<x<L,
 
</math>
 
</math>
 
</center>
 
</center>
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=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>
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<center>
 
<center>
 
<math>
 
<math>
\phi^{\mathrm{I}}  =e^{-k_{0}(x+L)}\phi_{0}\left(
+
\phi^{\mathrm{I}}  =e^{-k_{0}^{h}(x+L)}\phi_{0}\left(
 
z\right)  
 
z\right)  
 
</math>
 
</math>
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<math>
 
<math>
 
\phi(x,z)=e^{-k_{0}^h (x+L)}\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<-L
+
z\right) + \sum_{m=0}^{\infty}a_{m}e^{k_{m}^h (x+L)}\phi_{m}^h(z), \;\;x<-L
 
</math>
 
</math>
 
</center>
 
</center>
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\phi(x,z)= \sum_{m=0}^{\infty}b_{m}
 
\phi(x,z)= \sum_{m=0}^{\infty}b_{m}
 
e^{-k_{m}^d (x+L)}\phi_{m}^d(z)
 
e^{-k_{m}^d (x+L)}\phi_{m}^d(z)
+ \sum_{m=0}^{\infty}c{m}
+
+ \sum_{m=0}^{\infty}c_{m}
 
e^{k_{m}^d (x-L)}\phi_{m}^d(z)
 
e^{k_{m}^d (x-L)}\phi_{m}^d(z)
 
, \;\;-d<z<0,\,\,-L<x<L
 
, \;\;-d<z<0,\,\,-L<x<L
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<center>
 
<center>
 
<math>
 
<math>
\phi(x,z)= \sum_{m=0}^{\infty}d_{m}
+
\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^{-\kappa_{m} (x+L)}\psi_{m}(z)
+\sum_{m=0}^{\infty}e_{m}
+
+ e_0 \frac{x+L}{2 L} +
e^{-\kappa_{m} (x-L)}\psi_{m}(z)
+
\sum_{m=1}^{\infty}e_{m}
, \;\;-h<z<-d,\,\,-L<x>L
+
e^{\kappa_{m} (x-L)}\psi_{m}(z)
 +
, \;\;-h<z<-d,\,\,-L<x<L
 
</math>
 
</math>
 
</center>
 
</center>
 
<center>
 
<center>
 
<math>
 
<math>
\phi(x,z)= \sum_{m=0}^{\infty}f_{m}e^{-k_{m}^h (x-L)}\phi_{m}^h(z), \;\;x>L
+
\phi(x,z)= \sum_{m=0}^{\infty}f_{m}e^{-k_{m}^h (x-L)}\phi_{m}^h(z), \;\;L<x
 
</math>
 
</math>
 
</center>
 
</center>
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The definition of all terms can be found in [[Eigenfunction Matching for Submerged Semi-Infinite Dock]],
 
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.
 
as can the solution method and the method to extend the solution to waves incident at an angle.
 
  
 
= Matlab Code =
 
= Matlab Code =
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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