Eigenfunction Matching for a Circular Dock - Revision history

Meylan at 23:59, 16 October 2009

2009-10-16T23:59:30Z

← Older revision		Revision as of 23:59, 16 October 2009
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			{{complete pages}}

	== Introduction ==		== Introduction ==

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

2008-09-27T09:01:11Z

- An infinite dimensional system of equations =

← Older revision		Revision as of 09:01, 27 September 2008
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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

Meylan at 08:59, 27 September 2008

2008-09-27T08:59:49Z

← Older revision		Revision as of 08:59, 27 September 2008
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	=Introduction=		== Introduction ==

	We show here a solution for a dock on [[Finite Depth]] water, which is circular. This is the three-dimensional		We show here a solution for a dock on [[Finite Depth]] water, which is circular. This is the three-dimensional
	analog of the [[Eigenfunction Matching for a Semi-Infinite Dock]].		analog of the [[Eigenfunction Matching for a Semi-Infinite 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>.

	= A Simple Method To Calculate The [[Diffraction Transfer Matrix]] For The Case Of A Circular Dock =		== A Simple Method To Calculate The [[Diffraction Transfer Matrix]] For The Case Of A Circular Dock ==

	Let's consider an incident wave whose potential has the following expression		Let's consider an incident wave whose potential has the following expression
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	matrix for the [[Diffraction Transfer Matrix]].		matrix for the [[Diffraction Transfer Matrix]].

	= 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
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	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 at 11:26, 8 September 2008

2008-09-08T11:26:35Z

← Older revision		Revision as of 11:26, 8 September 2008
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	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: /* Solution Method */

2008-08-26T05:44:59Z

Solution Method

← Older revision		Revision as of 05:44, 26 August 2008
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	=Solution Method=		=Solution Method=

	We ~~begin by separating~~ variables in the ~~vertical direction~~.		We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the two regions, <math>r<a</math>
			and <math>r>a</math>.

	{{separation of variables in cylindrical coordinates in finite depth}}		{{separation of variables in cylindrical coordinates in finite depth}}

Meylan: /* Solution Method */

2008-08-26T05:20:45Z

Solution Method

← Older revision		Revision as of 05:20, 26 August 2008
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	{{separation of variables for a dock}}		{{separation of variables for a dock}}

	~~break~~		{{free surface dock relations}}


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

	Since the solution must be bounded		Since the solution must be bounded
	we know that under the plate the solution will be a linear combination of		we know that under the plate the solution will be a linear combination of

Meylan: /* Solution Method */

2008-08-26T05:14:03Z

Solution Method

← Older revision		Revision as of 05:14, 26 August 2008
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	{{separation of variables in cylindrical coordinates in finite depth}}		{{separation of variables in cylindrical coordinates in finite depth}}

			{{separation of variables for a free surface}}

	{{separation of variables for a dock}}		{{separation of variables for a dock}}

			break

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

Meylan: /* Solution Method */

2008-08-26T05:12:55Z

Solution Method

← Older revision		Revision as of 05:12, 26 August 2008
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	We begin by separating variables in the vertical direction.		We begin by separating variables in the vertical direction.

			{{separation of variables in cylindrical coordinates in finite depth}}

	{{separation of variables for a dock}}		{{separation of variables for a dock}}

Meylan: /* Additional code */

2008-08-18T07:58:26Z

Additional code

← Older revision		Revision as of 07:58, 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:Problems with Cylindrical Symmetry]]		[[Category:Problems with Cylindrical Symmetry]]

Meylan: /* Solution Method */

2008-08-08T23:48:22Z

Solution Method

@@ Line 41: / Line 41: @@
 {{separation of variables for a dock}}
-We now solve for the function <math>\rho_{n}(r)</math>.
+{{separation of variables in cylindrical coordinates}}
-Using Laplace's equation in polar coordinates we obtain
+Since the solution must be bounded
 we know that under the plate the solution will be a linear combination of
 <math>I_{n}(y)</math> while outside the plate the solution will be a