Eigenfunction Matching for a Circular Floating Elastic Plate - Revision history

Meylan: Reverted edits by Waexu (Talk) to last revision by Meylan

2010-07-19T04:44:41Z

Reverted edits by Waexu (Talk) to last revision by Meylan

← Older revision		Revision as of 04:44, 19 July 2010
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	[[Category:Floating Elastic Plate]]		[[Category:Floating Elastic Plate]]
	[[Category:Eigenfunction Matching Method]]		[[Category:Eigenfunction Matching Method]]

	~~[http://www.besttermpaper.com essays term papers]~~

Waexu at 16:57, 28 June 2010

2010-06-28T16:57:44Z

← Older revision		Revision as of 16:57, 28 June 2010
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	[[Category:Floating Elastic Plate]]		[[Category:Floating Elastic Plate]]
	[[Category:Eigenfunction Matching Method]]		[[Category:Eigenfunction Matching Method]]

			[http://www.besttermpaper.com essays term papers]

Meylan at 00:06, 17 October 2009

2009-10-17T00:06:26Z

← Older revision		Revision as of 00:06, 17 October 2009
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	=Introduction=		{{complete pages}}

			==Introduction==

	We show here a solution for a [[Floating Elastic Plate]] on [[Finite Depth]] water		We show here a solution for a [[Floating Elastic Plate]] on [[Finite Depth]] water
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	The solution is an extension of the [[Eigenfunction Matching for a Circular Dock]].		The solution is an extension of the [[Eigenfunction Matching for a Circular Dock]].

	=Governing Equations=		==Governing Equations==

	We begin with the [[Frequency Domain Problem]] for a [[Floating Elastic Plate]]		We begin with the [[Frequency Domain Problem]] for a [[Floating Elastic Plate]]
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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}}

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

	Since the solution must be bounded		Since the solution must be bounded
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	</center>		</center>

	== Displacement of the plate ==		=== Displacement of the plate ===

	The surface displacement and the water velocity potential at		The surface displacement and the water velocity potential at
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	</center>		</center>

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

	The boundary conditions (3) and		The boundary conditions (3) and
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	coefficients of the water velocity potential in the plate covered region.		coefficients of the water velocity potential in the plate covered region.

	=Numerical Solution=		==Numerical Solution==

	To solve the system of equations (10) together		To solve the system of equations (10) together
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	this which are discussed in more detail in [[Zilman_Miloh 2000a\|Zilman and Miloh 2000]].		this which are discussed in more detail in [[Zilman_Miloh 2000a\|Zilman and Miloh 2000]].

	=The [[:Category:Shallow Depth\|Shallow Depth]] Theory of [[Zilman_Miloh_2000a\|Zilman and Miloh 2000]]=		==The [[:Category:Shallow Depth\|Shallow Depth]] Theory of [[Zilman_Miloh_2000a\|Zilman and Miloh 2000]]==

	The shallow water theory of [[Zilman_Miloh_2000a\|Zilman and Miloh 2000]] can be recovered by		The shallow water theory of [[Zilman_Miloh_2000a\|Zilman and Miloh 2000]] can be recovered by
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	one in the open water) exactly as for the theory of [[Zilman_Miloh_2000a\|Zilman and Miloh 2000]].		one in the open water) exactly as for the theory of [[Zilman_Miloh_2000a\|Zilman and Miloh 2000]].

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

	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 plate problems can be found here		A program to calculate the coefficients for circular plate 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
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	[[Category:Floating Elastic Plate]]		[[Category:Floating Elastic Plate]]
	[[Category:Eigenfunction Matching Method]]		[[Category:Eigenfunction Matching Method]]
	~~[[Category:Complete Pages]]~~

Meylan: /* The Shallow Depth Theory of Zilman and Miloh 2000 */

2009-08-12T01:43:43Z

The Shallow Depth Theory of Zilman and Miloh 2000

← Older revision		Revision as of 01:43, 12 August 2009
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	this which are discussed in more detail in [[Zilman_Miloh 2000a\|Zilman and Miloh 2000]].		this which are discussed in more detail in [[Zilman_Miloh 2000a\|Zilman and Miloh 2000]].

	=The [[Shallow Depth]] Theory of [[Zilman_Miloh_2000a\|Zilman and Miloh 2000]]=		=The [[:Category:Shallow Depth\|Shallow Depth]] Theory of [[Zilman_Miloh_2000a\|Zilman and Miloh 2000]]=

	The shallow water theory of [[Zilman_Miloh_2000a\|Zilman and Miloh 2000]] can be recovered by		The shallow water theory of [[Zilman_Miloh_2000a\|Zilman and Miloh 2000]] can be recovered by

Meylan: /* Introduction */

2009-08-12T01:43:23Z

Introduction

← Older revision		Revision as of 01:43, 12 August 2009
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	We show here a solution for a [[Floating Elastic Plate]] on [[Finite Depth]] water		We show here a solution for a [[Floating Elastic Plate]] on [[Finite Depth]] water
	based on [[Peter_Meylan_Chung_2004a\|Peter, Meylan and Chung 2004]]. A solution		based on [[Peter_Meylan_Chung_2004a\|Peter, Meylan and Chung 2004]]. A solution
	for [[Shallow Depth]] was given in [[Zilman_Miloh_2000a\|Zilman and Miloh 2000]] and we will also show this.		for [[:Category:Shallow Depth\|Shallow Depth]] was given in [[Zilman_Miloh_2000a\|Zilman and Miloh 2000]] and we will also show this.
	The solution is an extension of the [[Eigenfunction Matching for a Circular Dock]].		The solution is an extension of the [[Eigenfunction Matching for a Circular Dock]].

Meylan: /* Additional code */

2008-10-28T02:43:50Z

Additional code

← Older revision		Revision as of 02:43, 28 October 2008
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	[[Category:Problems with Cylindrical Symmetry]]		[[Category:Problems with Cylindrical Symmetry]]
	[[Category:~~Linear Hydroelasticity~~]]		[[Category:Floating Elastic Plate]]
	[[Category:Eigenfunction Matching Method]]		[[Category:Eigenfunction Matching Method]]
	[[Category:Complete Pages]]		[[Category:Complete Pages]]

Meylan: /* Solution Method */

2008-08-26T05:55:23Z

Solution Method

← Older revision		Revision as of 05:55, 26 August 2008
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	{{incident plane wave in cylindrical coordinates}}		{{incident plane wave in cylindrical coordinates}}

			==Expansion of the potential ==

	Since the solution must be bounded		Since the solution must be bounded

Meylan: /* Solution Method */

2008-08-26T05:54:29Z

Solution Method

@@ Line 59: / Line 59: @@
 {{separation of variables in cylindrical coordinates}}
 {{incident plane wave in cylindrical coordinates}}
 ==Boundary conditions==

Meylan: /* Incident potential */

2008-08-26T05:48:16Z

Incident potential

← Older revision		Revision as of 05:48, 26 August 2008
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	~~==Incident potential==~~		{{incident plane wave in cylindrical coordinates}}

	~~The~~ incident ~~potential is a~~ wave ~~of amplitude <math>A</math>~~
	in ~~displacement travelling in the positive <math>x</math>-direction.~~
	~~The incident potential can therefore be written as~~
	~~<center>~~
	~~<math>~~
	~~\phi^{\mathrm{I~~}} ~~=\frac{A}{i\sqrt{\alpha}}e^{k_{0}x}\phi_{0}\left(~~
	~~z\right)~~
	~~=\sum\limits_{n=-\infty}^{\infty}e_{n}I_{n}(k_{0}r)\phi_{0}\left(z\right)~~
	~~e^{i n \theta}~~
	~~</math>~~
	~~</center>~~
	~~where <math>e_{n}=A/\left(i\sqrt{\alpha}\right)</math>~~
	~~(we retain the dependence on <math>n</math> for situations~~
	~~where the incident potential might take another form).~~

	==Boundary conditions==		==Boundary conditions==

Meylan: /* Separation of variables */

2008-08-26T05:47:20Z

Separation of variables

← Older revision		Revision as of 05:47, 26 August 2008
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	{{separation of variables in cylindrical coordinates}}		{{separation of variables in cylindrical coordinates}}

	~~== Separation of variables==~~

	~~We now separate variables, noting that since the problem has~~
	~~circular symmetry we can write the potential as~~
	~~<center>~~
	~~<math>~~
	~~\phi(r,\theta,z)=\zeta(z)\sum_{n=-\infty}^{\infty}\rho_{n}(r)e^{i n \theta}~~
	~~</math>~~
	~~</center>~~
	~~Applying Laplace's equation we obtain~~
	~~<center>~~
	~~<math>~~
	~~\zeta_{zz}+\mu^{2}\zeta=0~~
	~~</math>~~
	~~</center>~~
	~~so that:~~
	~~<center>~~
	~~<math>~~
	~~\zeta=\cos\mu(z+H)~~
	~~</math>~~
	~~</center>~~
	~~where the separation constant <math>\mu^{2}</math> must~~
	~~satisfy the [[Dispersion Relation for a Free Surface]]~~
	~~<center>~~
	~~<math>~~
	~~k\tan\left( kH\right) =-\alpha,\quad r>a\,\,\,(1)~~
	~~</math>~~
	~~</center>~~
	~~and the [[Dispersion Relation for a Floating Elastic Plate]]~~
	~~<center>~~
	~~<math>~~
	~~\kappa\tan(\kappa H)=\frac{-\alpha}{\beta\kappa^{4}+1-\alpha\gamma},\quad~~
	~~r<a \,\,\,(2)~~
	~~</math>~~
	~~</center>~~
	~~Note that we have set <math>\mu=k</math> under the free~~
	~~surface and <math>\mu=\kappa</math> under the plate. We denote the~~
	~~positive imaginary solution of (1) by <math>k_{0}</math> and~~
	~~the positive real solutions by <math>k_{m}</math>, <math>m\geq1</math>. The solutions of~~
	~~(2) will be denoted by~~
	~~<math>\kappa_{m}</math>, <math>m\geq-2</math>. The fully complex~~
	~~solutions with positive imaginary part are <math>\kappa_{-2}</math> and~~
	~~<math>\kappa_{-1}</math> (where <math>\kappa_{-1}=\overline{\kappa_{-2}}</math>),~~
	~~the negative imaginary solution is <math>\kappa_{0}</math> and the positive real~~
	~~solutions are <math>\kappa_{m}</math>, <math>m\geq1</math>. We define~~
	~~<center>~~
	~~<math>~~
	~~\phi_{m}\left( z\right) =\frac{\cos k_{m}(z+H)}{\cos k_{m}H},\quad m\geq0~~
	~~</math>~~
	~~</center>~~
	~~as the vertical eigenfunction of the potential in the open~~
	~~water region and~~
	~~<center>~~
	~~<math>~~
	~~\psi_{m}\left( z\right) =\frac{\cos\kappa_{m}(z+H)}{\cos\kappa_{m}H},\quad~~
	~~m\geq-2~~
	~~</math>~~
	~~</center>~~
	~~as the vertical eigenfunction of the potential in the plate~~
	~~covered region. For later reference, we note that:~~
	~~<center>~~
	~~<math>~~
	~~\int\nolimits_{-H}^{0}\phi_{m}(z)\phi_{n}(z) d z=A_{m}\delta_{mn}~~
	~~</math>~~
	~~</center>~~
	~~where~~
	~~<center>~~
	~~<math>~~
	~~A_{m}=\frac{1}{2}\left( \frac{\cos k_{m}H\sin k_{m}H+k_{m}H}{k_{m}\cos~~
	~~^{2}k_{m}H}\right)~~
	~~</math>~~
	~~</center>~~
	~~and~~
	~~<center>~~
	~~<math>~~
	~~\int\nolimits_{-H}^{0}\phi_{n}(z)\psi_{m}(z) d z=B_{mn}~~
	~~</math>~~
	~~</center>~~
	~~where~~
	~~<center><math>~~
	~~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}~~
	~~^{2}-\kappa_{m}^{2}\right) }~~
	~~</math></center>~~

	~~We now solve for the function <math>\rho_{n}(r)</math>.~~
	~~Using Laplace's equation in polar coordinates we obtain~~
	~~<center>~~
	~~<math>~~
	~~\frac{\mathrm{d}^{2}\rho_{n}}{\mathrm{d}r^{2}}+\frac{1}{r}~~
	~~\frac{\mathrm{d}\rho_{n}}{\mathrm{d}r}-\left(~~
	~~\frac{n^{2}}{r^{2}}+\mu^{2}\right) \rho_{n}=0~~
	~~</math>~~
	~~</center>~~
	~~where <math>\mu</math> is <math>k_{m}</math> or~~
	~~<math>\kappa_{m},</math> depending on whether <math>r</math> is~~
	~~greater or less than <math>a</math>. We can convert this equation to the~~
	~~standard form by substituting <math>y=\mu r</math> to obtain~~
	~~<center>~~
	~~<math>~~
	~~y^{2}\frac{\mathrm{d}^{2}\rho_{n}}{\mathrm{d}y^{2}}+y\frac{\mathrm{d}\rho_{n}~~
	~~}{\rm{d}y}-(n^{2}+y^{2})\rho_{n}=0~~
	~~</math>~~
	~~</center>~~
	~~The solution of this equation is a linear combination of the~~
	~~modified Bessel functions of order <math>n</math>, <math>I_{n}(y)</math> and~~
	~~<math>K_{n}(y)</math> [[Abramowitz and Stegun 1964]]. 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~~
	~~linear combination of <math>K_{n}(y)</math>. 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}\sum_{m=-2}^{\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 potential==		==Incident potential==

← Older revision		Revision as of 01:43, 12 August 2009
Line 302:		Line 302:
	this which are discussed in more detail in [[Zilman_Miloh 2000a\|Zilman and Miloh 2000]].		this which are discussed in more detail in [[Zilman_Miloh 2000a\|Zilman and Miloh 2000]].

	=The [[Shallow Depth]] Theory of [[Zilman_Miloh_2000a\|Zilman and Miloh 2000]]=		=The [[:Category:Shallow Depth\|Shallow Depth]] Theory of [[Zilman_Miloh_2000a\|Zilman and Miloh 2000]]=

	The shallow water theory of [[Zilman_Miloh_2000a\|Zilman and Miloh 2000]] can be recovered by		The shallow water theory of [[Zilman_Miloh_2000a\|Zilman and Miloh 2000]] can be recovered by