Difference between revisions of "Two Identical Submerged Docks using Symmetry"
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− | = Introduction = | + | {{complete pages}} |
+ | |||
+ | == Introduction == | ||
This is the extension of [[Eigenfunction Matching for a Submerged Finite Dock]] using | This is the extension of [[Eigenfunction Matching for a Submerged Finite Dock]] using | ||
− | [[:Category:Symmetry in Two | + | [[:Category:Symmetry in Two Dimensions|Symmetry in Two Dimensions]].. 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 Semi-Infinite Dock]] and [[Two Identical Docks using Symmetry]] | [[Eigenfunction Matching for a Submerged Semi-Infinite Dock]] and [[Two Identical Docks using Symmetry]] | ||
− | =Governing Equations= | + | ==Governing Equations== |
We begin with the [[Frequency Domain Problem]] for the submerged dock in | We begin with the [[Frequency Domain Problem]] for the submerged dock in | ||
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<center> | <center> | ||
<math> | <math> | ||
− | \partial_z\phi=0, \,\, z=-d,\,- | + | \partial_z\phi=0, \,\, z=-d,\,-L_2<x<-L_1,\,\,{\rm and}\,\,L_1<x<L_2 |
</math> | </math> | ||
</center> | </center> | ||
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and a wave propagating away. | and a wave propagating away. | ||
− | =Solution Method= | + | ==Solution Method== |
We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the three regions, <math>x<0</math> | We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the three regions, <math>x<0</math> | ||
Line 49: | Line 51: | ||
<center> | <center> | ||
<math> | <math> | ||
− | \phi^{\mathrm{I}} =e^{- | + | \phi^{\mathrm{I}} =e^{-k^{h}_{0}(x+L_2)}\phi_{0}\left( |
z\right) | z\right) | ||
</math> | </math> | ||
</center> | </center> | ||
− | + | We use [[:Category:Symmetry in Two Dimensions|Symmetry in Two Dimensions]] and express the symmetric solution | |
− | + | as | |
<center> | <center> | ||
<math> | <math> | ||
− | \phi(x,z)=e^{-k_{0}^h (x+ | + | \phi(x,z)=e^{-k_{0}^h (x+L_2)}\phi_{0}^h\left( |
− | z\right) + \sum_{m=0}^{\infty}a_{m}e^{k_{m}^h x}\phi_{m}^h(z), \;\;x<- | + | z\right) + \sum_{m=0}^{\infty}a_{m}^{s}e^{k_{m}^h (x+L_2)}\phi_{m}^h(z), \;\;x<-L_2 |
</math> | </math> | ||
</center> | </center> | ||
<center> | <center> | ||
<math> | <math> | ||
− | \phi(x,z)= \sum_{m=0}^{\infty}b_{m} | + | \phi(x,z)= \sum_{m=0}^{\infty}b_{m}^{s} |
− | e^{-k_{m}^d (x+ | + | e^{-k_{m}^d (x+L_2)}\phi_{m}^d(z) |
− | + \sum_{m=0}^{\infty}c_{m} | + | + \sum_{m=0}^{\infty}c_{m}^{s} |
− | e^{k_{m}^d (x | + | e^{k_{m}^d (x+L_1)}\phi_{m}^d(z) |
− | , \;\;-d<z<0,\,\,- | + | , \;\;-d<z<0,\,\,x<-L_2,\,-L_1<x<L_1, {\rm or} \, x>L_2 |
</math> | </math> | ||
</center> | </center> | ||
Line 74: | Line 76: | ||
<center> | <center> | ||
<math> | <math> | ||
− | \phi(x,z)= \sum_{m= | + | \phi(x,z)= d_0^{s}\frac{x+L_1}{L_2-L_1} + \sum_{m=1}^{\infty}d_{m}^{s} |
− | e^{\kappa_{m} (x+ | + | e^{\kappa_{m} (x+L_2)}\psi_{m}(z) |
− | +\sum_{m=0}^{\infty}e_{m} | + | + |
− | e^{-\kappa_{m} (x | + | e_0^{s}\frac{x+L_2}{L_2-L_1} |
− | , \;\;-h<z<-d,\,\,- | + | \sum_{m=0}^{\infty}e_{m}^{s} |
+ | e^{-\kappa_{m} (x+L_1)}\psi_{m}(z) | ||
+ | , \;\;-h<z<-d,\,\,\,-L_2<x<-L_1, {\rm or} \, L_1<x<L_2, | ||
</math> | </math> | ||
</center> | </center> | ||
<center> | <center> | ||
<math> | <math> | ||
− | \phi(x,z)= \sum_{m=0}^{\infty}f_{m} | + | \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>L |
</math> | </math> | ||
</center> | </center> | ||
Line 90: | Line 94: | ||
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 = | + | The anti-symmetric solution is |
+ | <center> | ||
+ | <math> | ||
+ | \phi(x,z)=e^{-k_{0}^h (x+L_2)}\phi_{0}^h\left( | ||
+ | z\right) + \sum_{m=0}^{\infty}a_{m}^{a}e^{k_{m}^h (x+L_2)}\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_2)}\phi_{m}^d(z) | ||
+ | + \sum_{m=0}^{\infty}c_{m}^{a} | ||
+ | e^{k_{m}^d (x+L_1)}\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)= d_0^{a}\frac{x+L_1}{L_2-L_1} + \sum_{m=1}^{\infty}d_{m}^{a} | ||
+ | e^{-\kappa_{m} (x+L_2)}\psi_{m}(z) | ||
+ | + | ||
+ | e_0^{a}\frac{x+L_2}{L_2-L_1} | ||
+ | \sum_{m=1}^{\infty}e_{m}^{a} | ||
+ | e^{\kappa_{m} (x+L_1)}\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{\sinh(k_{m}^h x)}{\sinh(k_m^h L)}\phi_{m}^h(z), \;\;x>L | ||
+ | </math> | ||
+ | </center> | ||
+ | |||
+ | == Matlab Code == | ||
− | A program to calculate the coefficients for the submerged | + | A program to calculate the coefficients for the submerged two finite dock problem can be found here |
− | [http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/ | + | [http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/two_submerged_finite_docks_symmetry.m two_submerged_finite_docks_symmetry.m] |
− | == Additional code == | + | === Additional code === |
This program requires | This program requires | ||
− | + | * {{free surface dispersion equation code}} | |
− | + | ||
[[Category:Eigenfunction Matching Method]] | [[Category:Eigenfunction Matching Method]] | ||
[[Category:Pages with Matlab Code]] | [[Category:Pages with Matlab Code]] | ||
[[Category:Complete Pages]] | [[Category:Complete Pages]] | ||
+ | [[Category:Symmetry in Two Dimensions|Symmetry in Two Dimensions]] |
Latest revision as of 00:00, 17 October 2009
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=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^{h}_{0}(x+L_2)}\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_2)}\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty}a_{m}^{s}e^{k_{m}^h (x+L_2)}\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_2)}\phi_{m}^d(z) + \sum_{m=0}^{\infty}c_{m}^{s} e^{k_{m}^d (x+L_1)}\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)= d_0^{s}\frac{x+L_1}{L_2-L_1} + \sum_{m=1}^{\infty}d_{m}^{s} e^{\kappa_{m} (x+L_2)}\psi_{m}(z) + e_0^{s}\frac{x+L_2}{L_2-L_1} \sum_{m=0}^{\infty}e_{m}^{s} e^{-\kappa_{m} (x+L_1)}\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_2)}\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty}a_{m}^{a}e^{k_{m}^h (x+L_2)}\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_2)}\phi_{m}^d(z) + \sum_{m=0}^{\infty}c_{m}^{a} e^{k_{m}^d (x+L_1)}\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)= d_0^{a}\frac{x+L_1}{L_2-L_1} + \sum_{m=1}^{\infty}d_{m}^{a} e^{-\kappa_{m} (x+L_2)}\psi_{m}(z) + e_0^{a}\frac{x+L_2}{L_2-L_1} \sum_{m=1}^{\infty}e_{m}^{a} e^{\kappa_{m} (x+L_1)}\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{\sinh(k_{m}^h x)}{\sinh(k_m^h L)}\phi_{m}^h(z), \;\;x\gt L }[/math]
Matlab Code
A program to calculate the coefficients for the submerged two finite dock problem can be found here two_submerged_finite_docks_symmetry.m
Additional code
This program requires