Eigenfunction Matching for a Submerged Circular Dock - Revision history

Meylan at 00:01, 17 October 2009

2009-10-17T00:01:22Z

← Older revision		Revision as of 00:01, 17 October 2009
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	= Introduction =		{{complete pages}}

			== Introduction ==

	We present here the theory for a submerged circular dock. The details of the method		We present here the theory for a submerged circular dock. The details of the method
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	[[Eigenfunction Matching for a Circular Dock]]		[[Eigenfunction Matching for a Circular Dock]]

	=Governing Equations=		==Governing Equations==

	We begin with the [[Frequency Domain Problem]].		We begin with the [[Frequency Domain Problem]].
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	denotes the derivative in <math>z</math>-direction.		denotes the derivative in <math>z</math>-direction.

	=Solution Method=		==Solution Method==

	We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>r<a</math>		We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>r<a</math>
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	{{incident plane wave in cylindrical coordinates}}		{{incident plane wave in cylindrical coordinates}}

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

	The potential and its derivative must be continuous across the		The potential and its derivative must be continuous across the
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	</center>		</center>

	= Numerical Solution =		== Numerical Solution ==

	To solve the system of equations we set the upper limit of <math>l</math> to		To solve the system of equations we set the upper limit of <math>l</math> to
	be <math>M</math>.		be <math>M</math>.

	= Matlab Code =		== Matlab Code ==

	A program to calculate the coefficients for circular dock problems can be found here		A program to calculate the coefficients for circular dock problems can be found here
	[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/circle_submerged_dock_matching_one_n.m circle_submerged_dock_matching_one_n.m]		[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/circle_submerged_dock_matching_one_n.m circle_submerged_dock_matching_one_n.m]
	Note that this problem solves only for a single n.		Note that this problem solves only for a single n.
	== Additional code ==		=== Additional code ===

	This program requires		This program requires

Meylan: /* Additional code */

2008-09-11T10:48:31Z

Additional code

← Older revision		Revision as of 10:48, 11 September 2008
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	[[Category:Eigenfunction Matching Method]]		[[Category:Eigenfunction Matching Method]]
	[[Category:Pages with Matlab Code]]		[[Category:Pages with Matlab Code]]
			[[Category:Complete Pages]]

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

2008-09-11T09:14:32Z

An infinite dimensional system of equations

← Older revision		Revision as of 09:14, 11 September 2008
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	k_{0}I_{n}^{\prime}(k_{0}a)\phi_{0}\left( z\right) +\sum		k_{0}I_{n}^{\prime}(k_{0}a)\phi_{0}\left( z\right) +\sum
	_{m=0}^{\infty} a_{mn}k_{m}K_{n}^{\prime}(k_{m}a)\phi_{m}\left( z\right)		_{m=0}^{\infty} a_{mn}k_{m}K_{n}^{\prime}(k_{m}a)\phi_{m}\left( z\right)
	=b_{1n} \frac{\|n\|}{a} \chi_{0}(z) +\sum_{m=0\neq 1}^{\infty}		=b_{1n} \frac{\|n\|}{a} \chi_{0}(z) +\sum_{m=0,m\neq 1}^{\infty}
	b_{mn}\kappa_{m}I_{n}^{\prime}(\kappa_{m}a)\chi_{m}(z)		b_{mn}\kappa_{m}I_{n}^{\prime}(\kappa_{m}a)\chi_{m}(z)
	</math>		</math>
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	<math>		<math>
	I_{n}(k_{0}a)A_{0}\delta_{0l}+a_{ln}K_{n}(k_{l}a)A_{l}		I_{n}(k_{0}a)A_{0}\delta_{0l}+a_{ln}K_{n}(k_{l}a)A_{l}
	=b_{1n}B_{0l} + \sum_{m=0\neq 1}^{\infty}b_{mn}I_{n}(\kappa_{m}a)B_{ml}		=b_{1n}B_{0l} + \sum_{m=0,m\neq 1}^{\infty}b_{mn}I_{n}(\kappa_{m}a)B_{ml}
	</math>		</math>
	</center>		</center>

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

2008-09-08T11:36:27Z

An infinite dimensional system of equations

← Older revision		Revision as of 11:36, 8 September 2008
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	I_{n}(k_{0}a)\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}		I_{n}(k_{0}a)\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
	a_{mn} K_{n}(k_{m}a)\phi_{m}\left( z\right)		a_{mn} K_{n}(k_{m}a)\phi_{m}\left( z\right)
	= b_{1n} \chi_{0}(z) +\sum_{m=0,m\neq 1}^{\infty}b_{mn}I_{n}(\kappa_{m}a)\~~psi_~~{m}(z)		= b_{1n} \chi_{0}(z) +\sum_{m=0,m\neq 1}^{\infty}b_{mn}I_{n}(\kappa_{m}a)\chi_{m}(z)
	</math>		</math>
	</center>		</center>
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	k_{0}I_{n}^{\prime}(k_{0}a)\phi_{0}\left( z\right) +\sum		k_{0}I_{n}^{\prime}(k_{0}a)\phi_{0}\left( z\right) +\sum
	_{m=0}^{\infty} a_{mn}k_{m}K_{n}^{\prime}(k_{m}a)\phi_{m}\left( z\right)		_{m=0}^{\infty} a_{mn}k_{m}K_{n}^{\prime}(k_{m}a)\phi_{m}\left( z\right)
	=b_{1n} \frac{\|n\|}{a} \~~psi_~~{0}(z) +\sum_{m=0\neq 1}^{\infty}		=b_{1n} \frac{\|n\|}{a} \chi_{0}(z) +\sum_{m=0\neq 1}^{\infty}
	b_{mn}\kappa_{m}I_{n}^{\prime}(\kappa_{m}a)\chi_{m}(z)		b_{mn}\kappa_{m}I_{n}^{\prime}(\kappa_{m}a)\chi_{m}(z)
	</math>		</math>
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	<math>		<math>
	I_{n}(k_{0}a)A_{0}\delta_{0l}+a_{ln}K_{n}(k_{l}a)A_{l}		I_{n}(k_{0}a)A_{0}\delta_{0l}+a_{ln}K_{n}(k_{l}a)A_{l}
	=b_{0n}B_{0l} + \sum_{m=1}^{\infty}b_{mn}I_{n}(\kappa_{m}a)B_{ml}		=b_{1n}B_{0l} + \sum_{m=0\neq 1}^{\infty}b_{mn}I_{n}(\kappa_{m}a)B_{ml}
	</math>		</math>
	</center>		</center>
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	k_{0}I_{n}^{\prime}(k_{0}a)A_{0}\delta_{0l}+a_{ln}k_{l}K_{n}^{\prime		k_{0}I_{n}^{\prime}(k_{0}a)A_{0}\delta_{0l}+a_{ln}k_{l}K_{n}^{\prime
	}(k_{l}a)A_{l}		}(k_{l}a)A_{l}
	= b_{0n}B_{0l}\frac{\|n\|}{a} +		= b_{1n}B_{0l}\frac{\|n\|}{a} +
	\sum_{m=1}^{\infty}b_{mn}\kappa_{m}I_{n}^{\prime}(\kappa_{m}a)B_{ml}		\sum_{m=0,m\neq 1}^{\infty}b_{mn}\kappa_{m}I_{n}^{\prime}(\kappa_{m}a)B_{ml}
	</math>		</math>
	</center>		</center>

Meylan at 11:32, 8 September 2008

2008-09-08T11:32:48Z

← Older revision		Revision as of 11:32, 8 September 2008
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	{{free surface submerged plate relations}}		{{free surface submerged plate relations}}

	~~{{separation of variables in cylindrical coordinates}}~~

	{{separation of variables in cylindrical coordinates}}		{{separation of variables in cylindrical coordinates}}
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	<math>I_{n}(y)</math> while outside the plate the solution will be a		<math>I_{n}(y)</math> while outside the plate the solution will be a
	linear combination of <math>K_{n}(y)</math>.		linear combination of <math>K_{n}(y)</math>.
	The case <math>\~~kappa_0~~ =0</math> is a special case and the solution under		The case <math>\mu_1 =0</math> is a special case and the solution under
	the dock is <math>(r/a)^{\|n\|}</math>.		the dock is <math>(r/a)^{\|n\|}</math>.





	Therefore the potential can		Therefore the potential can
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	<center>		<center>
	<math>		<math>
	\phi(r,\theta,z)=\sum_{n=-\infty}^{\infty}b_{0n}(r/a)^{\|n\|} e^{i n\theta}\~~psi_~~{0}(z)+		\phi(r,\theta,z)=\sum_{n=-\infty}^{\infty}b_{1n}(r/a)^{\|n\|} e^{i n\theta}\chi_{1}(z)+
	\sum_{n=-\infty}^{\infty}\sum_{m=1}^{\infty}b_{mn}		\sum_{n=-\infty}^{\infty}\sum_{m=0,m\neq 1}^{\infty}b_{mn}
	I_{n}(\kappa_{m}r)e^{i n\theta}\~~psi_~~{m}(z), \;\;r<a		I_{n}(\kappa_{m}r)e^{i n\theta}\chi_{m}(z), \;\;r<a
	</math>		</math>
	</center>		</center>
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	I_{n}(k_{0}a)\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}		I_{n}(k_{0}a)\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
	a_{mn} K_{n}(k_{m}a)\phi_{m}\left( z\right)		a_{mn} K_{n}(k_{m}a)\phi_{m}\left( z\right)
	= b_{0n} \~~psi_~~{0}(z) +\sum_{m=1}^{\infty}b_{mn}I_{n}(\kappa_{m}a)\psi_{m}(z)		= b_{1n} \chi_{0}(z) +\sum_{m=0,m\neq 1}^{\infty}b_{mn}I_{n}(\kappa_{m}a)\psi_{m}(z)
	</math>		</math>
	</center>		</center>
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	k_{0}I_{n}^{\prime}(k_{0}a)\phi_{0}\left( z\right) +\sum		k_{0}I_{n}^{\prime}(k_{0}a)\phi_{0}\left( z\right) +\sum
	_{m=0}^{\infty} a_{mn}k_{m}K_{n}^{\prime}(k_{m}a)\phi_{m}\left( z\right)		_{m=0}^{\infty} a_{mn}k_{m}K_{n}^{\prime}(k_{m}a)\phi_{m}\left( z\right)
	=b_{0n} \frac{\|n\|}{a} \psi_{0}(z) +\sum_{m=1}^{\infty}		=b_{1n} \frac{\|n\|}{a} \psi_{0}(z) +\sum_{m=0\neq 1}^{\infty}
	b_{mn}\kappa_{m}I_{n}^{\prime}(\kappa_{m}a)\~~psi~~		b_{mn}\kappa_{m}I_{n}^{\prime}(\kappa_{m}a)\chi_{m}(z)
	_{m}(z)
	</math>		</math>
	</center>		</center>
	for each <math>n</math>.		for each <math>n</math>.
	We solve these equations by multiplying both equations by		We solve these equations by multiplying both equations by
	<math>\phi_{l}(z)</math> and integrating from <math>-H</math> to <math>0</math> to obtain:		<math>\phi_{l}(z)</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
	<center>		<center>
	<math>		<math>

Meylan: /* Introduction */

2008-09-08T11:27:31Z

Introduction

← Older revision		Revision as of 11:27, 8 September 2008
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	= Introduction =		= Introduction =

	We present here ~~very briefly~~ the theory for a submerged circular dock. The details of the method		We present here the theory for a submerged circular dock. The details of the method
	can be found in [[Eigenfunction Matching for a Submerged Semi-Infinite Dock]] and		can be found in [[Eigenfunction Matching for a Submerged Semi-Infinite Dock]] and
	[[Eigenfunction Matching for a Circular Dock]]		[[Eigenfunction Matching for a Circular Dock]]

Meylan: /* Matlab Code */

2008-09-08T11:24:49Z

Matlab Code

← Older revision		Revision as of 11:24, 8 September 2008
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			Therefore the potential can
			be expanded as
			<center>
			<math>
			\phi(r,\theta,z)=\sum_{n=-\infty}^{\infty}\sum_{m=0}^{\infty}a_{mn}K_{n}
			(k_{m}r)e^{i n\theta}\phi_{m}(z), \;\;r>a
			</math>
			</center>
			and
			<center>
			<math>
			\phi(r,\theta,z)=\sum_{n=-\infty}^{\infty}b_{0n}(r/a)^{\|n\|} e^{i n\theta}\psi_{0}(z)+
			\sum_{n=-\infty}^{\infty}\sum_{m=1}^{\infty}b_{mn}
			I_{n}(\kappa_{m}r)e^{i n\theta}\psi_{m}(z), \;\;r<a
			</math>
			</center>
			where <math>a_{mn}</math> and <math>b_{mn}</math>
			are the coefficients of the potential in the open water and
			the plate covered region respectively.

			{{incident plane wave in cylindrical coordinates}}

			==An infinite dimensional system of equations==

			The potential and its derivative must be continuous across the
			transition from open water to the plate covered region. Therefore, the
			potentials and their derivatives at <math>r=a</math> have to be equal
			for each angle and we obtain
			<center>
			<math>
			I_{n}(k_{0}a)\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
			a_{mn} K_{n}(k_{m}a)\phi_{m}\left( z\right)
			= b_{0n} \psi_{0}(z) +\sum_{m=1}^{\infty}b_{mn}I_{n}(\kappa_{m}a)\psi_{m}(z)
			</math>
			</center>
			and
			<center>
			<math>
			k_{0}I_{n}^{\prime}(k_{0}a)\phi_{0}\left( z\right) +\sum
			_{m=0}^{\infty} a_{mn}k_{m}K_{n}^{\prime}(k_{m}a)\phi_{m}\left( z\right)
			=b_{0n} \frac{\|n\|}{a} \psi_{0}(z) +\sum_{m=1}^{\infty}
			b_{mn}\kappa_{m}I_{n}^{\prime}(\kappa_{m}a)\psi
			_{m}(z)
			</math>
			</center>
			for each <math>n</math>.
			We solve these equations by multiplying both equations by
			<math>\phi_{l}(z)</math> and integrating from <math>-H</math> to <math>0</math> to obtain:
			<center>
			<math>
			I_{n}(k_{0}a)A_{0}\delta_{0l}+a_{ln}K_{n}(k_{l}a)A_{l}
			=b_{0n}B_{0l} + \sum_{m=1}^{\infty}b_{mn}I_{n}(\kappa_{m}a)B_{ml}
			</math>
			</center>
			and
			<center>
			<math>
			k_{0}I_{n}^{\prime}(k_{0}a)A_{0}\delta_{0l}+a_{ln}k_{l}K_{n}^{\prime
			}(k_{l}a)A_{l}
			= b_{0n}B_{0l}\frac{\|n\|}{a} +
			\sum_{m=1}^{\infty}b_{mn}\kappa_{m}I_{n}^{\prime}(\kappa_{m}a)B_{ml}
			</math>
			</center>

			= Numerical Solution =

			To solve the system of equations we set the upper limit of <math>l</math> to
			be <math>M</math>.

	= Matlab Code =		= Matlab Code =

Meylan: /* Equations to solve */

2008-09-08T11:23:51Z

Equations to solve

← Older revision		Revision as of 11:23, 8 September 2008
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	~~= Equations to solve =~~

	~~<center>~~
	~~<math>~~
	~~I_{n}(k_{0}^{h}a)A_{0}\delta_{0l}+a_{ln}K_{n}(k_{l}^{h}a)A_{l}~~
	~~=\sum_{m=0}^{\infty}b_{mn}I_{n}(k_{m}^{\prime}a)B_{ml}^{\prime}~~
	~~</math>~~
	~~</center>~~
	~~and~~
	~~<center>~~
	~~<math>~~
	~~k_{0}I_{n}^{\prime}(k_{0}^{h}a)A_{0}\delta_{0l}+a_{ln}k_{l}^{h}K_{n}^{\prime~~
	~~}(k_{l}a)A_{l}~~
	~~= \sum_{m=0}^{\infty}b_{mn}k_{m}^{\prime}I_{n}^{\prime}(k_{m}^{\prime}a)B_{ml}^{\prime}~~
	~~</math>~~
	~~</center>~~
	~~where the definition of <math>A_{l}</math> and <math>B_{ml}^{\prime}</math> can be found in~~
	~~[[Eigenfunction Matching for a Submerged Semi-Infinite Dock]]~~

	~~To solve the system of equations we set the upper limit of <math>l</math> to~~
	~~be <math>M</math>.~~

	= Matlab Code =		= Matlab Code =

Meylan: /* Separation of Variables */

2008-09-08T11:23:02Z

Separation of Variables

← Older revision		Revision as of 11:23, 8 September 2008
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	the dock is <math>(r/a)^{\|n\|}</math>.		the dock is <math>(r/a)^{\|n\|}</math>.

	~~= Separation of Variables =~~

	~~We can separate variables and we obtain the following expression for the velocity potential~~

	~~Therefore the potential can~~
	~~be expanded as~~
	~~<center>~~
	~~<math>~~
	~~\phi(r,\theta,z)=\sum_{n=-\infty}^{\infty}\sum_{m=0}^{\infty}a_{mn}K_{n}~~
	~~(k_{m}^{h}r)e^{i n\theta}\phi^{h}_{m}(z), \;\;r>a~~
	~~</math>~~
	~~</center>~~
	~~and~~
	~~<center>~~
	~~<math>~~
	~~\phi(r,\theta,z)=~~
	~~\sum_{n=-\infty}^{\infty}\sum_{m=0}^{\infty}b_{mn}~~
	~~I_{n}(k_{m}^{\prime}r)e^{i n\theta}\chi_{m}(z), \;\;r<a~~
	~~</math>~~
	~~</center>~~
	~~where the definition of <math>k_{m}^{h}</math>, <math>k_{m}^{\prime}</math>, <math>\phi^{h}_{m}</math> and <math>\chi_{m}(z)</math>~~
	~~can be found in [[Eigenfunction Matching for a Submerged Semi-Infinite Dock]] and the expansion in the~~
	~~cylindrical eigenfunctions can be found in [[Eigenfunction Matching for a Circular Dock]]~~

	= Equations to solve =		= Equations to solve =

Meylan: /* Separation of Variables */

2008-09-08T11:21:29Z

Separation of Variables

@@ Line 35: / Line 35: @@
 as <math>r\rightarrow\infty</math>. The subscript <math>z</math>
 denotes the derivative in <math>z</math>-direction.
 = Separation of Variables =