Two Identical Submerged Docks using Symmetry - Revision history

Meylan at 00:00, 17 October 2009

2009-10-17T00:00:41Z

← Older revision		Revision as of 00:00, 17 October 2009
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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
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	[[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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	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>
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	</center>		</center>

	= Matlab Code =		== Matlab Code ==

	A program to calculate the coefficients for the submerged two finite dock problem can be found here		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/two_submerged_finite_docks_symmetry.m two_submerged_finite_docks_symmetry.m]		[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

Meylan: /* Solution Method */

2009-08-14T00:23:29Z

Solution Method

← Older revision		Revision as of 00:23, 14 August 2009
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	<center>		<center>
	<math>		<math>
	\phi^{\mathrm{I}} =e^{-k_{0}(x+L_2)}\phi_{0}\left(		\phi^{\mathrm{I}} =e^{-k^{h}_{0}(x+L_2)}\phi_{0}\left(
	z\right)		z\right)
	</math>		</math>
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	<math>		<math>
	\phi(x,z)= d_0^{a}\frac{x+L_1}{L_2-L_1} + \sum_{m=1}^{\infty}d_{m}^{a}		\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^{-\kappa_{m} (x+L_2)}\psi_{m}(z)
	+		+
	e_0^{a}\frac{x+L_2}{L_2-L_1}		e_0^{a}\frac{x+L_2}{L_2-L_1}
	\sum_{m=0}^{\infty}e_{m}^{a}		\sum_{m=1}^{\infty}e_{m}^{a}
	e^{-\kappa_{m} (x+L_1)}\psi_{m}(z)		e^{\kappa_{m} (x+L_1)}\psi_{m}(z)
	, \;\;-h<z<-d,\,\,\,-L_2<x<-L_1, {\rm or} \, L_1<x<L_2,		, \;\;-h<z<-d,\,\,\,-L_2<x<-L_1, {\rm or} \, L_1<x<L_2,
	</math>		</math>

Meylan: /* Additional code */

2008-09-16T10:35:08Z

Additional code

← Older revision		Revision as of 10:35, 16 September 2008
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	[[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]]

Meylan: /* Solution Method */

2008-08-26T08:12:57Z

Solution Method

← Older revision		Revision as of 08:12, 26 August 2008
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	<center>		<center>
	<math>		<math>
	\phi(x,z)= d_0\frac{x+L_1}{L_2-L_1} + \sum_{m=1}^{\infty}d_{m}^{s}		\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^{\kappa_{m} (x+L_2)}\psi_{m}(z)
	+		+
	e_0\frac{x+L_2}{L_2-L_1}		e_0^{s}\frac{x+L_2}{L_2-L_1}
	\sum_{m=0}^{\infty}e_{m}^{s}		\sum_{m=0}^{\infty}e_{m}^{s}
	e^{-\kappa_{m} (x+L_1)}\psi_{m}(z)		e^{-\kappa_{m} (x+L_1)}\psi_{m}(z)
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	<center>		<center>
	<math>		<math>
	\phi(x,z)=e^{-k_{0}^h (x+L)}\phi_{0}^h\left(		\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}\phi_{m}^h(z), \;\;x<-L_2		z\right) + \sum_{m=0}^{\infty}a_{m}^{a}e^{k_{m}^h (x+L_2)}\phi_{m}^h(z), \;\;x<-L_2
	</math>		</math>
	</center>		</center>
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	<math>		<math>
	\phi(x,z)= \sum_{m=0}^{\infty}b_{m}^{a}		\phi(x,z)= \sum_{m=0}^{\infty}b_{m}^{a}
	e^{-k_{m}^d (x+L)}\phi_{m}^d(z)		e^{-k_{m}^d (x+L_2)}\phi_{m}^d(z)
	+ \sum_{m=0}^{\infty}c_{m}^{a}		+ \sum_{m=0}^{\infty}c_{m}^{a}
	e^{k_{m}^d (x-L)}\phi_{m}^d(z)		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		, \;\;-d<z<0,\,\,x<-L_2,\,-L_1<x<L_1, {\rm or} \, x>L_2
	</math>		</math>
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	<center>		<center>
	<math>		<math>
	\phi(x,z)= \sum_{m=0}^{\infty}d_{m}^{a}		\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)}\psi_{m}(z)		e^{\kappa_{m} (x+L_2)}\psi_{m}(z)
	+\sum_{m=0}^{\infty}e_{m}^{a}		+
	e^{-\kappa_{m} (x-L)}\psi_{m}(z)		e_0^{a}\frac{x+L_2}{L_2-L_1}
			\sum_{m=0}^{\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,		, \;\;-h<z<-d,\,\,\,-L_2<x<-L_1, {\rm or} \, L_1<x<L_2,
	</math>		</math>
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	<center>		<center>
	<math>		<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), \;\;~~-L_1<~~x<0		\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>		</math>
	</center>		</center>

Meylan: /* Solution Method */

2008-08-26T08:10:59Z

Solution Method

← Older revision		Revision as of 08:10, 26 August 2008
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	<center>		<center>
	<math>		<math>
	\phi(x,z)= \sum_{m=0}^{\infty}d_{m}^{s}		\phi(x,z)= d_0\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^{\kappa_{m} (x+L_2)}\psi_{m}(z)
	+\sum_{m=0}^{\infty}e_{m}^{s}		+
			e_0\frac{x+L_2}{L_2-L_1}
			\sum_{m=0}^{\infty}e_{m}^{s}
	e^{-\kappa_{m} (x+L_1)}\psi_{m}(z)		e^{-\kappa_{m} (x+L_1)}\psi_{m}(z)
	, \;\;-h<z<-d,\,\,\,-L_2<x<-L_1, {\rm or} \, L_1<x<L_2,		, \;\;-h<z<-d,\,\,\,-L_2<x<-L_1, {\rm or} \, L_1<x<L_2,

Meylan: /* Solution Method */

2008-08-26T08:08:05Z

Solution Method

← Older revision		Revision as of 08:08, 26 August 2008
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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}(x+L_2)}\phi_{0}\left(
	z\right)		z\right)
	</math>		</math>
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	<center>		<center>
	<math>		<math>
	\phi(x,z)=e^{-k_{0}^h (x+L)}\phi_{0}^h\left(		\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}\phi_{m}^h(z), \;\;x<-L_2		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>
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	<math>		<math>
	\phi(x,z)= \sum_{m=0}^{\infty}b_{m}^{s}		\phi(x,z)= \sum_{m=0}^{\infty}b_{m}^{s}
	e^{-k_{m}^d (x+L)}\phi_{m}^d(z)		e^{-k_{m}^d (x+L_2)}\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)		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		, \;\;-d<z<0,\,\,x<-L_2,\,-L_1<x<L_1, {\rm or} \, x>L_2
	</math>		</math>
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	<math>		<math>
	\phi(x,z)= \sum_{m=0}^{\infty}d_{m}^{s}		\phi(x,z)= \sum_{m=0}^{\infty}d_{m}^{s}
	e^{\kappa_{m} (x+L)}\psi_{m}(z)		e^{\kappa_{m} (x+L_2)}\psi_{m}(z)
	+\sum_{m=0}^{\infty}e_{m}^{s}		+\sum_{m=0}^{\infty}e_{m}^{s}
	e^{-\kappa_{m} (x-L)}\psi_{m}(z)		e^{-\kappa_{m} (x+L_1)}\psi_{m}(z)
	, \;\;-h<z<-d,\,\,\,-L_2<x<-L_1, {\rm or} \, L_1<x<L_2,		, \;\;-h<z<-d,\,\,\,-L_2<x<-L_1, {\rm or} \, L_1<x<L_2,
	</math>		</math>

Meylan: /* Solution Method */

2008-08-20T01:48:47Z

Solution Method

← Older revision		Revision as of 01:48, 20 August 2008
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	<center>		<center>
	<math>		<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), \;\;-L1<x<0		\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), \;\;-L_1<x<0
	</math>		</math>
	</center>		</center>

Meylan: /* Matlab Code */

2008-08-18T08:08:03Z

Matlab Code

← Older revision		Revision as of 08:08, 18 August 2008
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	= 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 two finite dock problem can be found here
	[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/~~submerged_finite_dock~~.m ~~submerged_finite_dock~~.m]		[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 ==

Meylan: /* Additional code */

2008-08-18T08:03:25Z

Additional code

← Older revision		Revision as of 08:03, 18 August 2008
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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]]

Meylan: /* Solution Method */

2008-08-04T10:52:13Z

Solution Method

← Older revision		Revision as of 10:52, 4 August 2008
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	<center>		<center>
	<math>		<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		\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), \;\;-L1<x<0
	</math>		</math>
	</center>		</center>