Eigenfunction Matching for a Finite Dock - Revision history

Meylan at 02:11, 7 April 2010

2010-04-07T02:11:22Z

@@ Line 13: / Line 13: @@
 using [[:Category:Symmetry in Two Dimensions|Symmetry in Two Dimensions]] to two docks of the same length and this can be
 found [[Two Identical Docks using Symmetry]]. We also show how the solution
-can be found using [[:Category:Symmetry in Two Dimensions|Symmetry in Two Dimensions]] for the finite dock.
+can be found using [[:Category:Symmetry in Two Dimensions|Symmetry in Two Dimensions]] for the finite dock
 ==Governing Equations==
@@ Line 208: / Line 209: @@
 * {{free surface dispersion equation code}}
 [[Category:Eigenfunction Matching Method]]
 [[Category:Pages with Matlab Code]]

Meylan at 00:02, 17 September 2009

2009-09-17T00:02:38Z

← Older revision		Revision as of 00:02, 17 September 2009
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			{{complete pages}}

	== Introduction ==		== Introduction ==

Meylan: /* Governing Equations */

2009-09-16T23:59:14Z

Governing Equations

@@ Line 15: / Line 15: @@
 ==Governing Equations==
-We consider here the [[Frequency Domain Problem]] for a finite dock which occupies
+{{finite dock equations}}
 ==Solution Method==

Meylan: /* Introduction */

2009-09-16T23:58:25Z

Introduction

← Older revision		Revision as of 23:58, 16 September 2009
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	using [[:Category:Symmetry in Two Dimensions\|Symmetry in Two Dimensions]] to two docks of the same length and this can be		using [[:Category:Symmetry in Two Dimensions\|Symmetry in Two Dimensions]] to two docks of the same length and this can be
	found [[Two Identical Docks using Symmetry]]. We also show how the solution		found [[Two Identical Docks using Symmetry]]. We also show how the solution
	can be found using [[:Category:Symmetry in Two Dimensions\|Symmetry in Two Dimensions]] for the finite dock.		can be found using [[:Category:Symmetry in Two Dimensions\|Symmetry in Two Dimensions]] for the finite dock.

	~~[[Image:finite_dock.jpg\|thumb\|right\|300px\|Wave scattering by a finite dock]]~~

	==Governing Equations==		==Governing Equations==

Meylan: /* Matlab Code */

2008-09-27T08:55:59Z

Matlab Code

← Older revision		Revision as of 08:55, 27 September 2008
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	</center>		</center>

	=== 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

Adi Kurniawan at 04:18, 23 September 2008

2008-09-23T04:18:00Z

← Older revision		Revision as of 04:18, 23 September 2008
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	= Introduction =		== Introduction ==

	The problems consists of a region to the left and right		The problems consists of a region to the left and right
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	[[Image:finite_dock.jpg\|thumb\|right\|300px\|Wave scattering by a finite dock]]		[[Image:finite_dock.jpg\|thumb\|right\|300px\|Wave scattering by a finite dock]]

	=Governing Equations=		==Governing Equations==

	We consider here the [[Frequency Domain Problem]] for a finite dock which occupies		We consider here the [[Frequency Domain Problem]] for a finite dock which occupies
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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, exactly		We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables] in the three regions, exactly
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	{{incident potential for two dimensions}}		{{incident potential for two dimensions}}

	= Expansion of the Potential =		== Expansion of the Potential ==
	The potential can		The potential can
	be expanded as		be expanded as
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	covered region. We have an incident wave from the left.		covered region. We have an incident wave from the left.

	==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.

	= 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>.		We can consider the problem when the waves are incident at an angle <math>\theta</math>.
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	and these are solved exactly as before.		and these are solved exactly as before.

	= 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/finite_dock.m finite_dock.m]		[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/finite_dock.m finite_dock.m]

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

	This program requires		This program requires
	* {{free surface dispersion equation code}}		* {{free surface dispersion equation code}}

	= Solution using Symmetry =		== Solution using Symmetry ==

	The finite dock problem is symmetric about the line <math>x=0</math> and this allows us to solve the problem		The finite dock problem is symmetric about the line <math>x=0</math> and this allows us to solve the problem
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	[[:Category:Symmetry in Two Dimensions\|Symmetry in Two Dimensions]]		[[:Category:Symmetry in Two Dimensions\|Symmetry in Two Dimensions]]

	== Symmetric solution ==		=== Symmetric solution ===
	The symmetric potential can		The symmetric potential can
	be expanded as		be expanded as
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	== Anti-~~Symmetric~~ solution ==		=== Anti-symmetric solution ===

	The anti-symmetric potential can		The anti-symmetric potential can
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	(for full details of this derivation see [[Eigenfunction Matching for a Semi-Infinite Dock]])		(for full details of this derivation see [[Eigenfunction Matching for a Semi-Infinite Dock]])

	== Solution to the original problem ==		=== Solution to the original problem ===

	We can now reconstruct the potential for the finite dock from the two previous symmetric and anti-symmetric solution as explained in		We can now reconstruct the potential for the finite dock from the two previous symmetric and anti-symmetric solution as explained in
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	</center>		</center>

	== 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/finite_dock_symmetry.m finite_dock_symmetry.m]		[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/finite_dock_symmetry.m finite_dock_symmetry.m]

	== Additional Code ==		=== Additional Code ===
	This program requires		This program requires
	* {{free surface dispersion equation code}}		* {{free surface dispersion equation code}}

Meylan: /* Solution Method */

2008-08-26T04:51:23Z

Solution Method

← Older revision		Revision as of 04:51, 26 August 2008
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	{{free surface dock relations}}		{{free surface dock relations}}

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

	= Expansion of the Potential =		= Expansion of the Potential =

Meylan: /* Solution Method */

2008-08-26T04:50:10Z

Solution Method

← Older revision		Revision as of 04:50, 26 August 2008
Line 58:		Line 58:
	{{free surface dock relations}}		{{free surface dock relations}}

	{{incident ~~wave~~}}		{{incident potential two dimensions}}

	= Expansion of the Potential =		= Expansion of the Potential =

Meylan: /* Expansion of the Potential */

2008-08-26T04:48:08Z

Expansion of the Potential

← Older revision		Revision as of 04:48, 26 August 2008
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	{{free surface dock relations}}		{{free surface dock relations}}

			{{incident wave}}

	= Expansion of the Potential =		= Expansion of the Potential =

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

2008-08-26T04:47:04Z

Solution with Waves Incident at an Angle

← Older revision		Revision as of 04:47, 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>.
	~~case we have the wavenumber in the <math>y</math> direction is <math>k_y = \sin\theta k_0</math>~~		{{incident angle}}
	~~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~~
	~~under the dock.~~

	~~This means that the potential is now of the form <math>\phi(x,y,z)=e^~~{~~k_y y~~}~~\phi(x,z)</math>~~
	~~so that when we separate variables we obtain~~

	Therefore the potential can		Therefore the potential can