Green Function Methods for Floating Elastic Plates - Revision history

Meylan: /* Equation in Terms of the Modes of the Plate */

2010-04-28T10:50:57Z

Equation in Terms of the Modes of the Plate

← Older revision		Revision as of 10:50, 28 April 2010
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	\omega^2 a_{mn}(\omega) -i\omega b_{mn}(\omega) = - i\omega\int_{-L}^{L}\phi_m^{\mathrm{R}}X_{n}\mathrm{d}x,		\omega^2 a_{mn}(\omega) -i\omega b_{mn}(\omega) = - i\omega\int_{-L}^{L}\phi_m^{\mathrm{R}}X_{n}\mathrm{d}x,
	</math></center>		</math></center>
	and they are referred to as the added mass and damping coefficients (see [[Linear Wave-Body ~~Interactions~~]]		and they are referred to as the added mass and damping coefficients (see [[Linear Wave-Body Interaction]]
	for the equivalent definition for a rigid body).		for the equivalent definition for a rigid body).
	respectively.		respectively.

2010-04-28T10:50:25Z

Equation in Terms of the Modes of the Plate

← Older revision		Revision as of 10:50, 28 April 2010
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	\omega^2 a_{mn}(\omega) -i\omega b_{mn}(\omega) = - i\omega\int_{-L}^{L}\phi_m^{\mathrm{R}}X_{n}\mathrm{d}x,		\omega^2 a_{mn}(\omega) -i\omega b_{mn}(\omega) = - i\omega\int_{-L}^{L}\phi_m^{\mathrm{R}}X_{n}\mathrm{d}x,
	</math></center>		</math></center>
	and they are referred to as the added mass and damping coefficients (see [[~~Added~~-~~Mass, Damping Coefficients And Exciting Forces~~]]		and they are referred to as the added mass and damping coefficients (see [[Linear Wave-Body Interactions]]
	for the equivalent definition for a rigid body).		for the equivalent definition for a rigid body).
	respectively.		respectively.

2010-04-28T10:40:55Z

Transformation using Eigenfunctions for a Uniform Free Beam

← Older revision		Revision as of 10:40, 28 April 2010
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	<center>		<center>
	<math>		<math>
	\frac{\partial \phi}{\partial z} = i\omega \sum_{n=0}^{\infty} \zeta_n X_n		\frac{\partial \phi}{\partial z} = -\mathrm{i}\omega \sum_{n=0}^{\infty} \zeta_n X_n
	</math>		</math>
	</center>		</center>

2009-11-26T03:24:23Z

Equation in Terms of the Modes of the Plate

← Older revision		Revision as of 03:24, 26 November 2009
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	As the plate is floating on the surface, we can denote it as follows:		As the plate is floating on the surface, we can denote it as follows:
	<center><math>		<center><math>
	\phi^{\rm I}\|_{z=0} = e^{-~~i kx~~} \,		\phi^{\rm I}\|_{z=0} = e^{-k_0 x} \,
	</math></center>		</math></center>
			where we have set the amplitude to be unity.

	We now consider the scattered potentials <math>\phi^{\mathrm{S}}</math>. The relationship between scattered potentials, diffracted potentials and the incident wave are as follows:		We now consider the scattered potentials <math>\phi^{\mathrm{S}}</math>. The relationship between scattered potentials, diffracted potentials and the incident wave are as follows:

2009-09-21T01:18:57Z

Solution for the Radiation and Diffracted Potential

← Older revision		Revision as of 01:18, 21 September 2009
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	and		and
	~~<center><math>~~
	~~\phi_n^~~{~~\mathrm~~{R}}~~(x) = \int_{-L}^{L}G(x,\xi)~~		{{Green's function equations for the radiation potential for a dock like structure}}
	~~\left(~~
	~~\alpha\phi_n^{\mathrm{R}}(\xi) - i\omega X_n(\xi)~~
	~~\right)\mathrm{d} \xi~~
	~~</math></center>~~

	== Reflection and Transmission Coefficients ==		== Reflection and Transmission Coefficients ==

2009-09-21T01:13:45Z

Equation in Terms of the Modes of the Plate

← Older revision		Revision as of 01:13, 21 September 2009
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	which satisfy the following equations		which satisfy the following equations

	{{radiation ~~conditions~~ for a dock like structure}}		{{radiation potential equations for a dock like structure}}

	Therefore we find the potential as		Therefore we find the potential as

2009-09-21T01:11:37Z

← Older revision		Revision as of 01:11, 21 September 2009
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	</math></center>		</math></center>
	which satisfy the following equations		which satisfy the following equations
	~~<center><math>~~
	~~\Delta\phi_n^~~{~~\mathrm~~{~~R}} =0,\,\,-h<z<0,~~		{{radiation conditions for a dock like structure}}
	~~</math></center>~~
	~~<center><math>~~
	~~\partial_{z}\phi_n^{\mathrm{R}} =0,\,\,z=-h,~~
	~~</math></center>~~
	~~<center><math>~~
	~~\partial_{z}\phi_n^{\mathrm{R}} =\alpha\phi_n^{\mathrm{R}},\,\,x\notin(-L,L),\, \,~~
	~~z=0~~
	~~</math></center>~~
	~~<center><math>~~
	~~\partial_{z}\phi_n^{\mathrm{R}} = i\omega X_{n},\,\,x\in(-L,L),\,\,z=0.~~
	~~</math></center>~~
	~~The~~ radiation ~~condition~~ for ~~the radiation potential is~~
	~~<center><math>~~
	~~\frac{\partial\phi_n^{\mathrm{R~~}}~~}{\partial x}\pm ik\phi_n^{\mathrm{R}}=0,\,\,\mathrm{as}~~
	~~\,\,x\rightarrow\pm\infty.~~
	~~</math></center>~~
	Therefore we find the potential as		Therefore we find the potential as
	<center><math>		<center><math>

2009-09-17T00:37:05Z

Matlab Code

← Older revision		Revision as of 00:37, 17 September 2009
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	== Matlab Code ==		== Matlab Code ==


	~~A program to calculate the solution in elastic modes can be found here~~
	{{elastic plate modes code}}		{{elastic plate modes code}}

	~~=== Additional code ===~~

	~~This program requires~~
	* {{green function surface code}}
	* {{green function source and field free surface code}}
	* {{free surface dispersion equation code}}
	* {{eigenvalues beam}}
	* {{eigenvectors beam}}

	== Alternative Solution Method using Green Functions for a Uniform Plate ==		== Alternative Solution Method using Green Functions for a Uniform Plate ==

2009-09-17T00:14:00Z

Reflection and Transmission Coefficients

@@ Line 150: / Line 150: @@
 == Reflection and Transmission Coefficients ==
-The Reflection and Transmission Coefficients represent the ratio of the amplitude of the reflected or transmitted wave to the amplitude of the incident wave.  They hold the property that <math> |R|^2+|T|^2=1\,</math> (and may often contain an imaginery element).
+{{derivation of reflection and transmission in two dimensions}}
 == Matlab Code ==

2009-09-17T00:13:11Z

Solution for the Radiation and Diffracted Potential

← Older revision		Revision as of 00:13, 17 September 2009
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	== Solution for the Radiation and Diffracted Potential ==		== Solution for the Radiation and Diffracted Potential ==

	~~We use the [[Free-Surface~~ Green ~~Function]]~~ for ~~two-dimensional waves, with singularity at~~		{{Green's function equations for the diffracted potential}}
	~~the water surface since we are only~~
	~~interested in its value at <math>z=0.</math> Using this we can transform~~ the ~~system of equations to~~

	~~<center><math>~~
	~~\phi^{\mathrm{D}}(x) = \phi^{\mathrm{I}}(x) + \int_{-L}^{L}G(x,\xi) \alpha\phi^{\mathrm{D}}(\xi) \mathrm{d} \xi~~
	~~</math></center>~~
	and		and
	<center><math>		<center><math>
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	\right)\mathrm{d} \xi		\right)\mathrm{d} \xi
	</math></center>		</math></center>

	~~Details about this method can be found in [[Integral Equation for the Finite Depth Green Function at Surface]].~~

	== Reflection and Transmission Coefficients ==		== Reflection and Transmission Coefficients ==