Green Function Method for a Finite Dock - Revision history

Meylan at 00:42, 17 September 2009

2009-09-17T00:42:37Z

← Older revision		Revision as of 00:42, 17 September 2009
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	{{~~incomplete~~ pages}}		{{complete pages}}

	== Introduction ==		== Introduction ==

Meylan: /* Introduction */

2009-09-17T00:41:14Z

Introduction

← Older revision		Revision as of 00:41, 17 September 2009
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	The problem of a two-dimensional finite dock is solved using a green function.		The problem of a two-dimensional finite dock is solved using a green function.
	The same problem is solved using eigenfunction matching in [[Eigenfunction Matching for a Finite Dock]].		The same problem is solved using eigenfunction matching in [[Eigenfunction Matching for a Finite Dock]].
			The problem of a floating body on the surface using the same method is treated in
			[[Green Function Method for a Floating Body on the Surface]] and for a floating elastic plate
			in [[Green Function Methods for Floating Elastic Plates]]

	== Equations for a Dock in the Frequency Domain ==		== Equations for a Dock in the Frequency Domain ==

Meylan: /* Matlab Code */

2009-09-17T00:38:00Z

Matlab Code

← Older revision		Revision as of 00:38, 17 September 2009
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	== Matlab Code ==		== Matlab Code ==

	{{~~elastic plate modes~~ code}}		{{finite dock green function code}}


	[[Category:Linear Water-Wave Theory]]		[[Category:Linear Water-Wave Theory]]

Meylan: /* Matlab Code */

2009-09-17T00:36:11Z

Matlab Code

← Older revision		Revision as of 00:36, 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}}


	[[Category:Linear Water-Wave Theory]]		[[Category:Linear Water-Wave Theory]]

Meylan at 00:18, 17 September 2009

2009-09-17T00:18:36Z

← Older revision		Revision as of 00:18, 17 September 2009
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			[[Category:Linear Water-Wave Theory]]
	~~== Solution for the Radiation and Diffracted Potential ==~~

	~~We use the~~ [[~~Free-Surface Green Function]] for two-dimensional waves, with singularity at~~
	~~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{S}}(\xi) \mathrm{d} \xi~~
	~~</math></center>~~
	~~and~~
	~~<center><math>~~
	~~\phi_n^{\mathrm{R}}(x) = \int_{-L}^{L}G(x,\xi)~~
	~~\left(~~
	~~\alpha\phi_n^{\mathrm{R}}(\xi) - i\omega X_n(\xi)~~
	~~\right)\mathrm{d} \xi~~
	~~</math></center>~~

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

	~~== 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).

	[[Image:Square_volume.png\|600px\|right\|thumb\|frame\|A diagram depicting the area <math>\Omega\,</math> which is bounded by the rectangle <math>\partial \Omega \,</math>. The rectangle <math>\partial \Omega \,</math> is bounded by <math> -h \leq z \leq 0 \,</math> and <math>-\infty \leq x \leq \infty \,</math> or <math>-N \leq x \leq N\,</math>]]

	~~We can calculate the Reflection and Transmission coefficients as follows:~~
	~~Applying Green's theorem to <math>\phi\,</math> and <math>\phi^{\mathrm{I}}\,</math> gives:~~
	~~<center><math>~~
	~~0 = \iint_{\Omega}(\phi\nabla^2\phi^{\mathrm{I}} - \phi^{\mathrm{I}}\nabla^2\phi)\mathrm{d}x\mathrm{d}z~~
	~~= \int_{\partial\Omega}(\phi\frac{\partial\phi^{\rm I}}{\partial n} - \phi^{\rm I}\frac{\partial\phi}{\partial n})\mathrm{d}l,~~
	~~</math></center>~~
	~~<center><math>~~
	~~= \phi_0(0) \int_{-L}^{L} e^{k_0 x} \left(\alpha \phi(x) - \partial_n \phi(x)\right)\mathrm{d}x~~
	~~- 2k_0 R \int_{-h}^{0} \left(\phi_0(z)\right)^2 \mathrm{d}z.~~
	~~</math></center><br\>~~
	~~where <math> k_0 \,</math> is the first imaginery root of the dispersion equation and the incident wave is of the form: <math> \phi^I=\phi_0(z)e^{-ikx} \,</math><br\><br\>~~
	~~Therefore, in the case of a floating plate (where z=0)~~:
	~~<center><math>~~
	~~R = \frac{\int_{-L}^{L} e^{k_0 x} \left(\alpha \phi(x) - \partial_n \phi(x)\right)\mathrm{d}x }~~
	~~{2 k_0 \int_{-h}^{0} \left(\phi_0(z)\right)^2 \mathrm{d}z}.~~
	~~</math></center>~~
	~~and using a wave incident from the right we obtain~~
	~~<center><math>~~
	~~1 + T = \frac{\int_{-L}^{L} e^{-k_0 x} \left(\alpha \phi(x) - \partial_n \phi(x)\right)\mathrm{d}x }~~
	~~{2 k_0 \int_{-h}^{0} \left(\phi_0(z)\right)^2 \mathrm{d}z}.~~
	~~</math></center>~~

	~~== Matlab Code ==~~


	~~A program to calculate the solution in elastic modes can be found here~~
	~~{{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 ==~~

	~~We can also solve the equation by a closely related method which was given in~~
	~~[[Meylan and Squire 1994]].~~
	~~We can transform the equations to~~
	~~<center><math>~~
	~~\phi(x) = \phi^{\rm I}(x) + \int_{-L}^{L}G(x,\xi)~~
	~~\left(~~
	~~\alpha\phi(\xi) - \partial_z\phi(\xi)~~
	~~\right)\mathrm{d} \xi~~
	~~</math></center>~~

	~~Expanding as before~~
	~~<center>~~
	~~<math>~~
	~~\partial_z \phi = i\omega \sum \xi_n X_n~~
	~~</math>~~
	~~</center>~~
	~~we obtain~~
	~~<center><math>~~
	~~-i\omega \phi = \sum \left(\beta\lambda_n^4 - \gamma\alpha + 1\right)\xi_n X_n~~
	~~</math>~~
	~~</center>~~
	~~This leads to the following equation~~
	~~<center>~~
	~~<math>~~
	~~\partial_z\phi(x) = \frac{1}{\alpha} \int_{~~-~~L}^{L} \frac{X_n(x)X_n(\xi)}{\beta\lambda_n^4 - \gamma\alpha + 1} \phi(\xi)\mathrm{d}\xi~~
	~~</math>~~
	~~</center>~~
	or
	~~<center>~~
	~~<math>~~
	~~\partial_z\phi(x) = \frac{1}{\alpha} \int_{-L}^{L} g(x,\xi) \phi(\xi)\mathrm{d}\xi~~
	~~</math>~~
	~~</center>~~
	~~where~~
	~~<center>~~
	~~<math>~~
	~~g(x,\xi) = \frac{X_n(x)X_n(\xi)}{\beta\lambda_n^4 - \gamma\alpha + 1}~~
	~~</math>~~
	~~</center>~~
	~~which is the Green function for the plate.~~

	~~[[Category:Floating Elastic Plate~~]]

Meylan: /* Alternative Solution Method using Green Functions for a Uniform Plate */

2009-09-17T00:17:09Z

Alternative Solution Method using Green Functions for a Uniform Plate

Show changes

Meylan: /* Solution for the Radiation and Diffracted Potential */

2009-09-17T00:16:53Z

Solution for the Radiation and Diffracted Potential

← Older revision		Revision as of 00:16, 17 September 2009
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	</math></center>		</math></center>

	== Solution for ~~the Radiation and~~ Diffracted Potential ==		== Solution for Diffracted Potential ==

	{{Green's function equations for the diffracted potential}}		{{Green's function equations for the diffracted potential}}

	~~and~~
	~~<center><math>~~
	~~\phi_n^{\mathrm{R}}(x) = \int_{-L}^{L}G(x,\xi)~~
	~~\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 ==

Meylan: /* Equations for a Dock in the Frequency Domain */

2009-09-17T00:16:36Z

Equations for a Dock in the Frequency Domain

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 \phi^{\rm I}|_{z=0}  =  e^{-i kx} \,
 </math></center>
 == Solution for the Radiation and Diffracted Potential ==

Meylan: /* Equations for a Dock in the Frequency Domain */

2009-09-17T00:15:36Z

Equations for a Dock in the Frequency Domain

Show changes

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

2009-09-17T00:15:23Z

Equation in Terms of the Modes of the Plate

← Older revision		Revision as of 00:15, 17 September 2009
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	~~== Equation in Terms of the Modes of the Plate ==~~

	~~Under these assumptions, the equations become~~
	~~<center><math>~~
	~~\Delta\phi =0,\,\,-h<z<0,~~
	~~</math></center>~~
	~~<center><math>~~
	~~\partial_{z}\phi =0,\,\,z=-h,~~
	~~</math></center>~~
	~~<center><math>~~
	~~\alpha\phi =\partial_{z}\phi,\,\,x\notin(-L,L),\ \ z=0,~~
	~~</math></center>~~
	~~<center><math>~~
	~~i\omega\sum_{n=0}^{\infty}\zeta_{n}X_{n} =\partial_{z}\phi,\,\,x\in~~
	~~(-L,L),\,\, z=0,~~
	~~</math></center>~~
	~~<center><math>~~
	~~\sum_{n=0}^{\infty}\zeta_{n}\left( 1+\beta\lambda_{n}^{4}\right)~~
	~~X_{n}-\alpha\gamma\sum_{n=0}^{\infty}\zeta_{n}X_{n} = -i\omega~~
	~~\phi,\,\,x\in(-L,L),\,\, z=0.~~
	~~</math></center>~~
	~~We solve for the potential (and displacement) as the sum of~~
	~~the diffracted and radiation potentials in the standard way,~~
	~~as for a rigid body.~~
	~~<center><math>~~
	~~\phi=\phi^{\mathrm{D}}+\phi^{\mathrm{R}} ,\,~~
	~~</math></center>~~
	We begin with the diffraction potential <math>\phi^{\mathrm{D}}</math> which		We begin with the diffraction potential <math>\phi^{\mathrm{D}}</math> which
	satisfies the following equations		satisfies the following equations