Two Identical Docks using Symmetry - Revision history

Meylan: /* Matlab Code */

2010-02-14T23:29:40Z

Matlab Code

← Older revision		Revision as of 23:29, 14 February 2010
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	== Matlab Code ==		== Matlab Code ==

	~~A program to calculate the coefficients for the finite dock problems can be found here~~		{{two docks eigenfunction matching symmetry}}
	~~[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/two_finite_docks_symmetry.m two_finite_docks_symmetry.m]~~

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

Meylan at 23:59, 16 October 2009

2009-10-16T23:59:15Z

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

	== Introduction ==		== Introduction ==

Meylan at 08:57, 27 September 2008

2008-09-27T08:57:47Z

← Older revision		Revision as of 08:57, 27 September 2008
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	= Introduction =		== Introduction ==

	The problems consists three regions		The problems consists three regions
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	at an angle. For the later we give the equations in slightly less detail.		at an angle. For the later we give the equations in slightly less detail.

	=Governing Equations=		== Governing Equations ==

	We begin with the [[Frequency Domain Problem]] for a dock which occupies		We begin with the [[Frequency Domain Problem]] for a dock which occupies
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	and a wave propagating away.		and a wave propagating away.

	=Solution Method=		== Solution Method ==

	We begin by separating variables in the different regions.		We begin by separating variables in the different regions.
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	{{incident potential for two dimensions}}		{{incident potential for two dimensions}}

	== Expansion of the potential ==		=== Expansion of the potential ===
	The solution method uses [[:Category:Symmetry in Two Dimensions\|Symmetry in Two Dimensions]] and we write the potential as		The solution method uses [[:Category:Symmetry in Two Dimensions\|Symmetry in Two Dimensions]] and we write the potential as
	a symmetric and an anti-symmetric part and consider only the region <math>x<0</math>. We apply		a symmetric and an anti-symmetric part and consider only the region <math>x<0</math>. We apply
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	left and right and <math>b^s_m</math> and <math>c^s_m</math> are the coefficients under the dock.		left and right and <math>b^s_m</math> and <math>c^s_m</math> are the coefficients under the dock.

	==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>. We then simply need to solve the linear system of equations.		be <math>M</math>. We then simply need to solve the linear system of equations.

	= Anti-Symmetric Solution =		== Anti-Symmetric Solution ==

	The solution for the anti-symmetric potential proceeds in an almost identical manner.		The solution for the anti-symmetric potential proceeds in an almost identical manner.
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	</center>		</center>

	= Solution with Waves Incident at an Angle =		== Solution with Waves Incident at an Angle ==

	We can consider the problem when the waves are incident at an angle <math>\theta</math>. In some		We can consider the problem when the waves are incident at an angle <math>\theta</math>. In some
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	potential is found in a similar fashion.		potential is found in a similar fashion.

	= Matlab Code =		== Matlab Code ==

	A program to calculate the coefficients for the finite dock problems can be found here		A program to calculate the coefficients for the finite dock problems can be found here
	[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/two_finite_docks_symmetry.m two_finite_docks_symmetry.m]		[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/two_finite_docks_symmetry.m two_finite_docks_symmetry.m]

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

	This program requires		This program requires

Meylan: /* Solution with Waves Incident at an Angle */

2008-08-26T04:57:14Z

Solution with Waves Incident at an Angle

← Older revision		Revision as of 04:57, 26 August 2008
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	= Solution with Waves Incident at an Angle =		= Solution with Waves Incident at an Angle =

	We can consider the problem when the waves are incident at an angle <math>\theta</math>~~. In this~~		We can consider the problem when the waves are incident at an angle <math>\theta</math>. In some
	~~case we have the wavenumber in the <math>y</math> direction is <math>k_y = \sin\theta k_0</math>~~
	~~where <math>k_0</math> is as defined previously (note that <math>k_y</math> is imaginary)~~. In some
	ways the solution is now simpler because we do not need to write the zero term separately		ways the solution is now simpler because we do not need to write the zero term separately
	under the dock.		under the dock.
			{{incident angle}}

	This means that the potential is now of the form <math>\phi(x,y,z)=e^{k_y y}\phi(x,z)</math>.		This means that the potential is now of the form <math>\phi(x,y,z)=e^{k_y y}\phi(x,z)</math>.

Meylan: /* Additional code */

2008-08-26T04:56:02Z

Additional code

← Older revision		Revision as of 04:56, 26 August 2008
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	[[Category:Pages with Matlab Code]]		[[Category:Pages with Matlab Code]]
	[[Category:Symmetry in Two Dimensions]]		[[Category:Symmetry in Two Dimensions]]
			[[Category:Complete Pages]]

Meylan: /* Solution Method */

2008-08-26T04:55:35Z

Solution Method

← Older revision		Revision as of 04:55, 26 August 2008
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	We begin by separating variables in the different regions.		We begin by separating variables in the different regions.

			{{separation of variables in two dimensions}}

			{{separation of variables for a free surface}}

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

			{{free surface dock relations}}

			{{incident potential for two dimensions}}

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

Meylan: /* Additional code */

2008-08-18T08:01:56Z

Additional code

← Older revision		Revision as of 08:01, 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:Symmetry in Two Dimensions]]		[[Category:Symmetry in Two Dimensions]]

Meylan: /* Solution Method */

2008-08-08T22:20:36Z

Solution Method

@@ Line 46: / Line 46: @@
 =Solution Method=
 The solution method uses [[:Category:Symmetry in Two Dimensions|Symmetry in Two Dimensions]] and we write the potential as
 a symmetric and an anti-symmetric part and consider only the region <math>x<0</math>. We apply
@@ Line 76: / Line 81: @@
 are the coefficients of the potential in the open water regions to the
 left and right and <math>b^s_m</math> and <math>c^s_m</math> are the coefficients under the dock.
 ==An infinite dimensional system of equations==

Meylan: /* Solution Method */

2008-08-08T22:18:50Z

Solution Method

← Older revision		Revision as of 22:18, 8 August 2008
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	=Solution Method=		=Solution Method=

	The solution method uses [[Symmetry in Two Dimensions]] and we write the potential as		The solution method uses [[:Category:Symmetry in Two Dimensions\|Symmetry in Two Dimensions]] and we write the potential as
	a symmetric and an anti-symmetric part and consider only the region <math>x<0</math>. We apply		a symmetric and an anti-symmetric part and consider only the region <math>x<0</math>. We apply
	either Neuman (symmetric) or Dirichlet (anti-symmetric) boundary conditions at <math>x=0</math>.		either Neuman (symmetric) or Dirichlet (anti-symmetric) boundary conditions at <math>x=0</math>.

Meylan: /* Solution Method */

2008-07-11T09:11:37Z

Solution Method

← Older revision		Revision as of 09:11, 11 July 2008
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	<center><math>		<center><math>
	B_{mn}=\frac{k_{n}\sin k_{n}h\cos\kappa_{m}h-\kappa_{m}\cos k_{n}h\sin		B_{mn}=\frac{k_{n}\sin k_{n}h\cos\kappa_{m}h-\kappa_{m}\cos k_{n}h\sin
	\kappa_{m}h}{~~\left(~~ \cos k_{n}h~~\cos\kappa_{m}h\right)~~ \left( k_{n}		\kappa_{m}h}{ \cos k_{n}h \left( k_{n}
	^{2}-\kappa_{m}^{2}\right) }		^{2}-\kappa_{m}^{2}\right) }
	</math></center>		</math></center>