Difference between revisions of "Two Identical Submerged Docks using Symmetry"

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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}^{s}
 
+ \sum_{m=0}^{\infty}c_{m}^{s}
e^{k_{m}^d (x-L)}\phi_{m}^d(z)^{s}
+
e^{k_{m}^d (x-L)}\phi_{m}^d(z)
 
, \;\;-d<z<0,\,\,x<-L_2,\,-L_1<x<L_1, {\rm or} \, x>L_2
 
, \;\;-d<z<0,\,\,x<-L_2,\,-L_1<x<L_1, {\rm or} \, x>L_2
 
</math>
 
</math>
Line 89: Line 89:
 
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.
 +
 +
The anti-symmetric solution is
 +
<center>
 +
<math>
 +
\phi(x,z)=e^{-k_{0}^h (x+L)}\phi_{0}^h\left(
 +
z\right) + \sum_{m=0}^{\infty}a_{m}^{a}e^{k_{m}^h x}\phi_{m}^h(z), \;\;x<-L_2
 +
</math>
 +
</center>
 +
<center>
 +
<math>
 +
\phi(x,z)= \sum_{m=0}^{\infty}b_{m}^{a}
 +
e^{-k_{m}^d (x+L)}\phi_{m}^d(z)
 +
+ \sum_{m=0}^{\infty}c_{m}^{a}
 +
e^{k_{m}^d (x-L)}\phi_{m}^d(z)
 +
, \;\;-d<z<0,\,\,x<-L_2,\,-L_1<x<L_1, {\rm or} \, x>L_2
 +
</math>
 +
</center>
 +
and
 +
<center>
 +
<math>
 +
\phi(x,z)= \sum_{m=0}^{\infty}d_{m}^{a}
 +
e^{\kappa_{m} (x+L)}\psi_{m}(z)
 +
+\sum_{m=0}^{\infty}e_{m}^{a}
 +
e^{-\kappa_{m} (x-L)}\psi_{m}(z)
 +
, \;\;-h<z<-d,\,\,\,-L_2<x<-L_1, {\rm or} \, L_1<x<L_2,
 +
</math>
 +
</center>
 +
<center>
 +
<math>
 +
\phi(x,z)= \sum_{m=0}^{\infty}f_{m}^{a}\frac{\cosh(k_{m}^h x)}{\cosh(k_m^h L)}\phi_{m}^h(z), \;\;x>L
 +
</math>
 +
</center>
  
 
= Matlab Code =
 
= Matlab Code =

Revision as of 10:49, 4 August 2008

Introduction

This is the extension of Eigenfunction Matching for a Submerged Finite Dock using Symmetry in Two Dimensions.. 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 Two Identical Docks using Symmetry

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

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

[math]\displaystyle{ \partial_z\phi=0, \,\, z=-d,\,-L_2\lt x\lt -L_1,\,\,{\rm and}\,\,L_1\lt x\lt L_2 }[/math]

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

Solution Method

We use separation of variables in the three regions, [math]\displaystyle{ x\lt 0 }[/math] [math]\displaystyle{ -d\lt z\lt 0,\,\,x\gt 0 }[/math], and [math]\displaystyle{ -h\lt z\lt -d,\,\,x\gt 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]\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}(x+L)}\phi_{0}\left( z\right) }[/math]

We use Symmetry in Two Dimensions and express the symmetric solution as

[math]\displaystyle{ \phi(x,z)=e^{-k_{0}^h (x+L)}\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty}a_{m}^{s}e^{k_{m}^h x}\phi_{m}^h(z), \;\;x\lt -L_2 }[/math]

[math]\displaystyle{ \phi(x,z)= \sum_{m=0}^{\infty}b_{m}^{s} e^{-k_{m}^d (x+L)}\phi_{m}^d(z) + \sum_{m=0}^{\infty}c_{m}^{s} e^{k_{m}^d (x-L)}\phi_{m}^d(z) , \;\;-d\lt z\lt 0,\,\,x\lt -L_2,\,-L_1\lt x\lt L_1, {\rm or} \, x\gt L_2 }[/math]

and

[math]\displaystyle{ \phi(x,z)= \sum_{m=0}^{\infty}d_{m}^{s} e^{\kappa_{m} (x+L)}\psi_{m}(z) +\sum_{m=0}^{\infty}e_{m}^{s} e^{-\kappa_{m} (x-L)}\psi_{m}(z) , \;\;-h\lt z\lt -d,\,\,\,-L_2\lt x\lt -L_1, {\rm or} \, L_1\lt x\lt L_2, }[/math]

[math]\displaystyle{ \phi(x,z)= \sum_{m=0}^{\infty}f_{m}^{s}\frac{\cosh(k_{m}^h x)}{\cosh(k_m^h L)}\phi_{m}^h(z), \;\;x\gt L }[/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.

The anti-symmetric solution is

[math]\displaystyle{ \phi(x,z)=e^{-k_{0}^h (x+L)}\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty}a_{m}^{a}e^{k_{m}^h x}\phi_{m}^h(z), \;\;x\lt -L_2 }[/math]

[math]\displaystyle{ \phi(x,z)= \sum_{m=0}^{\infty}b_{m}^{a} e^{-k_{m}^d (x+L)}\phi_{m}^d(z) + \sum_{m=0}^{\infty}c_{m}^{a} e^{k_{m}^d (x-L)}\phi_{m}^d(z) , \;\;-d\lt z\lt 0,\,\,x\lt -L_2,\,-L_1\lt x\lt L_1, {\rm or} \, x\gt L_2 }[/math]

and

[math]\displaystyle{ \phi(x,z)= \sum_{m=0}^{\infty}d_{m}^{a} e^{\kappa_{m} (x+L)}\psi_{m}(z) +\sum_{m=0}^{\infty}e_{m}^{a} e^{-\kappa_{m} (x-L)}\psi_{m}(z) , \;\;-h\lt z\lt -d,\,\,\,-L_2\lt x\lt -L_1, {\rm or} \, L_1\lt x\lt L_2, }[/math]

[math]\displaystyle{ \phi(x,z)= \sum_{m=0}^{\infty}f_{m}^{a}\frac{\cosh(k_{m}^h x)}{\cosh(k_m^h L)}\phi_{m}^h(z), \;\;x\gt L }[/math]

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 dispersion_free_surface.m to run