Eigenfunction Matching for a Finite Change in Depth - Revision history

Meylan at 08:46, 21 June 2011

2011-06-21T08:46:58Z

← Older revision		Revision as of 08:46, 21 June 2011
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	{{~~incomplete~~ pages}}		{{complete pages}}

	== Introduction ==		== Introduction ==

Meylan: /* Matlab Code */

2011-06-21T08:45:44Z

Matlab Code

← Older revision		Revision as of 08:45, 21 June 2011
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	A program to calculate the coefficients for the semi-infinite dock problems can be found here		A program to calculate the coefficients for the semi-infinite dock problems can be found here
	{{~~semiinfinite_change_in_depth~~ code}}		{{finite_change_in_depth code}}

	=== Additional code ===		=== Additional code ===

Meylan: /* Inner product between free surface and dock modes */

2011-06-21T08:32:18Z

Inner product between free surface and dock modes

← Older revision		Revision as of 08:32, 21 June 2011
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	~~=== Inner product between free surface and dock modes ===~~

	~~<center><math>~~
	~~B_{mn} = \int\nolimits_{-h}^{0}\phi_{m}^{d}(z)\phi_{n}^{h}(z) \mathrm{d} z~~
	~~</math></center>~~
	~~where~~
	~~<center><math>~~
	~~B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(k_{m}^{h}(z+h))\cos(k_{n}^{d}(z+d))}{\cos(k_{m}h)\cos(k_{n}d)} \mathrm{d} z~~
	~~=\frac{k_{m}^{h}\sin(k_{m}^{d} h)\cos(k_{n}^{d}h)-k_{n}^{d}\cos(k_{m}^{h} h)\sin(k_{n}^{d}h)}~~
	~~{\cos(k_{m}h)\cos(k_{n}d)({k_{m}^{d}}^{2}-{k_{n}^{d}}^{2})}~~
	~~</math></center>~~

	== Matlab Code ==		== Matlab Code ==

Meylan: /* An infinite dimensional system of equations */

2011-06-21T08:31:04Z

An infinite dimensional system of equations

← Older revision		Revision as of 08:31, 21 June 2011
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	{{separation of variables for a free surface}}		{{separation of variables for a free surface}}

	~~=== An infinite dimensional system of equations ===~~


	~~The potential and its derivative must be continuous across <math>x=0</math>. Therefore, the~~
	~~potentials and their derivatives at <math>x=0</math> have to be equal or equal to zero~~
	~~as appropriate.~~
	~~We obtain:~~
	~~<center><math>~~
	~~\begin{align}~~
	~~\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), \quad -h<z<0 \\~~
	~~-\sum_{m=0}^{\infty}b_{m}k_{m}^{d}\phi_{m}^{d}(z)= &~~
	~~\begin{cases}~~
	~~-k_{0}^{h}\phi_{0}^{h}\left( z\right) +\sum_{m=0}^{\infty} a_{m}k_{m}^{h}\phi_{m}^{h}\left( z\right),\quad -h<z<0 \\~~
	~~0,\quad -d<z<-h~~
	~~\end{cases}~~
	~~\end{align}~~
	~~</math></center>~~

	~~For the first equation we multiply both sides by <math>\phi_{n}^{h}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:~~
	~~<center><math>~~
	~~A_{0}^{h}\delta_{0n} + a_{n}A_{n}^{h} = \sum^{\infty}_{m=0}b_{m}B_{mn}, n\in\mathbb{N}\cup\left\{0\right\}~~
	~~</math></center>~~
	~~and for the second equation we multiply both sides by <math>\phi_{n}^{d}(z) \,</math> and integrating from <math>-d</math> to <math>0</math> to obtain:~~
	~~<center><math>~~
	~~-k_{0}^{h}B_{n0} + \sum^{\infty}_{m=0}a_{m}k_{m}^{h}B_{nm} = -b_{n}^{d}k_{n}^{d}A_{n}^{d}, n\in\mathbb{N}\cup\left\{0\right\}~~
	~~</math></center>~~

	~~Solving the equations above will yield the coefficients of the water velocity potential in the dock covered region.~~

	=== Inner product between free surface and dock modes ===		=== Inner product between free surface and dock modes ===

Meylan: /* Expansion of the potential */

2011-06-21T08:30:44Z

Expansion of the potential

← Older revision		Revision as of 08:30, 21 June 2011
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	{{separation of variables for a free surface}}		{{separation of variables for a free surface}}

	~~=== Expansion of the potential ===~~

	~~We need to apply some boundary conditions at plus and minus infinity,~~
	~~where are essentially the the solution cannot grow. This means that we~~
	~~only have the positive (or negative) roots of the dispersion equation.~~
	~~However, it does not help us with the purely imaginary root. Here we~~
	~~must use a different condition, essentially identifying one solution~~
	~~as the incoming wave and the other as the outgoing wave.~~

	~~Therefore the scattered potential (without the incident wave, which will~~
	~~be added later) can~~
	~~be expanded as~~
	~~<center>~~
	~~<math>~~
	~~\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}^{h} x}\phi_{m}^{h}(z), \;\;x<0~~
	~~</math>~~
	~~</center>~~
	~~and~~
	~~<center>~~
	~~<math>~~
	~~\phi(x,z)=\sum_{m=0}^{\infty}b_{m}~~
	~~e^{-k_{m}^{d}x}\phi_{m}^{d}(z), \;\;x>0~~
	~~</math>~~
	~~</center>~~
	~~where <math>a_{m}</math> and <math>b_{m}</math>~~
	~~are the coefficients of the potential in the left and right respectively~~
	~~and <math>k_{m}^{h}</math> denotes the solution for depth <math>h</math> etc.~~

	~~{{incident potential for two dimensions}}~~

	=== An infinite dimensional system of equations ===		=== An infinite dimensional system of equations ===

Meylan: /* Matlab Code */

2011-06-21T08:30:19Z

Matlab Code

@@ Line 121: / Line 121: @@
 {\cos(k_{m}h)\cos(k_{n}d)({k_{m}^{d}}^{2}-{k_{n}^{d}}^{2})}
 </math></center>
 == Governing Equations ==

Meylan: /* Introduction */

2011-06-20T12:25:54Z

Introduction

← Older revision		Revision as of 12:25, 20 June 2011
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	== Introduction ==		== Introduction ==

	The problem consists of a region of free water surface with depth <math>d</math> except between		The problem consists of a region of free water surface with depth <math>h</math> except between
	<math>-L</math> and <math>L</math> where the depth is <math>h</math>.		<math>-L</math> and <math>L</math> where the depth is <math>d</math>.
	The problem with a semi-infinite change in depth is treated in		The problem with a semi-infinite change in depth is treated in
	[[Eigenfunction Matching for a Semi-Infinite Change in Depth]]		[[Eigenfunction Matching for a Semi-Infinite Change in Depth]]

Meylan: /* Governing Equations */

2011-06-20T12:25:14Z

Governing Equations

← Older revision		Revision as of 12:25, 20 June 2011
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	The boundary value problem can therefore be expressed as		The boundary value problem can therefore be expressed as
	<center><math>		<center><math>
	\Delta\phi=0, \,\, -h<z<0, \,\, x<0		\Delta\phi=0, \,\, -h<z<0, \,\, x<-L,\,x>L
	</math></center>		</math></center>
	<center><math>		<center><math>
	\Delta\phi=0, \,\, -d<z<0, \,\, x>0		\Delta\phi=0, \,\, -d<z<0, \,\, -L<x<L
	</math></center>		</math></center>
	<center><math>		<center><math>
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	</math></center>		</math></center>
	<center><math>		<center><math>
	\partial_x\phi=0, \,\, -d<z<-h,\,x=0,		\partial_x\phi=0, \,\, -d<z<-h,\,x=\pm L,
	</math></center>		</math></center>
	<center><math>		<center><math>
	\partial_z\phi=0, \,\, z=-h,\, x<0		\partial_z\phi=0, \,\, z=-h,\, x<-L,\,x>L
	</math></center>		</math></center>
	<center><math>		<center><math>
	\partial_z\phi=0, \,\, z=-d,\, x>0		\partial_z\phi=0, \,\, z=-d,\, -L<x<L
	</math></center>		</math></center>

Meylan at 12:18, 20 June 2011

2011-06-20T12:18:14Z

@@ Line 3: / Line 3: @@
 == Introduction ==
-The problem consists of a region to the left with a free water surface with depth <math>d</math>
+The problem consists of a region of free water surface with depth <math>d</math> except between
-and to the right a region of depth <math>h</math>. The problem with a semi-infinite change in depth is treated in
+<math>-L</math> and <math>L</math> where the depth is <math>h</math>.
 [[Eigenfunction Matching for a Semi-Infinite Change in Depth]]
 == Governing Equations ==

Meylan: /* Symmetric solution */

2010-10-13T23:18:46Z

Symmetric solution

← Older revision		Revision as of 23:18, 13 October 2010
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	<math>		<math>
	-k_{0}^{h}B_{n0} + \sum_{m=0}^{\infty} a_{m}^{s}k_{m}^{h} B_{nm}		-k_{0}^{h}B_{n0} + \sum_{m=0}^{\infty} a_{m}^{s}k_{m}^{h} B_{nm}
	= b_{n}^{s}k_{n}^{d}\tanh(k_{n}^{d}L) A_{n}^{d}		= -b_{n}^{s}k_{n}^{d}\tanh(k_{n}^{d}L) A_{n}^{d}
	</math>		</math>
	</center>		</center>