Green Function Solution Method - Revision history

Meylan at 19:34, 8 February 2010

2010-02-08T19:34:06Z

2010-02-08T19:33:54Z

Standard Linear Wave Scattering Problem

← Older revision		Revision as of 19:33, 8 February 2010
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	We then use [http://en.wikipedia.org/wiki/Green's_identities Green's second identity]		We then use [http://en.wikipedia.org/wiki/Green's_identities Green's second identity]
	If φ and ψ are both twice continuously differentiable on ~~''U''~~, then		If φ and ψ are both twice continuously differentiable on <math>\Omega</math>, then
	<center><math> \~~int_U~~ \left( \psi \nabla^2 \varphi - \varphi \nabla^2 \psi\right)\, dV =		<center><math> \int_\Omega \left( \psi \nabla^2 \varphi - \varphi \nabla^2 \psi\right)\, dV =
	\oint_{\partial U} \left( \psi {\partial \varphi \over \partial n} - \varphi {\partial \psi \over \partial n}\right)\, dS		\oint_{\partial \Omega} \left( \psi {\partial \varphi \over \partial n} - \varphi {\partial \psi \over \partial n}\right)\, dS
	</math></center>		</math></center>
	If we then substitute the [[Free-Surface Green Function]] which satisfies the following equations (plus the		If we then substitute the [[Free-Surface Green Function]] which satisfies the following equations (plus the

2010-02-08T19:29:51Z

2010-02-08T19:27:35Z

2009-11-06T13:19:11Z

Standard Linear Wave Scattering Problem

← Older revision		Revision as of 13:19, 6 November 2009
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	<center>		<center>
	<math>		<math>
	\frac{\partial G}{\partial z} = \alpha ~~\phi~~,\,z=0.		\frac{\partial G}{\partial z} = \alpha G,\,z=0.
	</math>		</math>
	</center>		</center>

2009-11-06T10:47:50Z

Standard Linear Wave Scattering Problem

← Older revision		Revision as of 10:47, 6 November 2009
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	for ψ we obtain		for ψ we obtain
	<center><math>		<center><math>
	\phi^{in} +		\phi^\mathrm{I} +
	\int_{\partial \Omega }\left(		\int_{\partial \Omega }\left(
	G_{n}\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi \left( \mathbf{x}		G_{n}\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi \left( \mathbf{x}
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	\left(		\left(
	\begin{matrix}		\begin{matrix}
	0, \,\,\,x\notin U \cup \partial U, \\		0, \,\,\,x\notin \Omega \cup \partial \Omega, \\
	\phi(\mathbf{x})/2,\,\,\,\mathbf{x} \in \partial U, \\		\phi(\mathbf{x})/2,\,\,\,\mathbf{x} \in \partial \Omega, \\
	\phi(\mathbf{x}),\,\,\,\mathbf{x} \in U,		\phi(\mathbf{x}),\,\,\,\mathbf{x} \in \Omega,
	\end{matrix}		\end{matrix}
	\right.		\right.

2009-11-05T09:33:57Z

← Older revision		Revision as of 09:33, 5 November 2009
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	\oint_{\partial U} \left( \psi {\partial \varphi \over \partial n} - \varphi {\partial \psi \over \partial n}\right)\, dS		\oint_{\partial U} \left( \psi {\partial \varphi \over \partial n} - \varphi {\partial \psi \over \partial n}\right)\, dS
	</math></center>		</math></center>
	If we then ~~substitiute~~ the [[Free-Surface Green Function]] which satisfies the following equations (plus the		If we then substitute the [[Free-Surface Green Function]] which satisfies the following equations (plus the
	[[Sommerfeld Radiation Condition]] far from the body)		[[Sommerfeld Radiation Condition]] far from the body)
	<center>		<center>

2009-11-05T09:31:46Z

← Older revision		Revision as of 09:31, 5 November 2009
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	<center>		<center>
	<math>		<math>
	\nabla_{\mathbf{x}}^{2}G(\mathbf{x},\mathbf{\xi})=\delta(\mathbf{x}-\mathbf{\xi}), \, -~~\infty~~<z<0		\nabla_{\mathbf{x}}^{2}G(\mathbf{x},\mathbf{\xi})=\delta(\mathbf{x}-\mathbf{\xi}), \, -h<z<0
	</math>		</math>
	</center>		</center>

2008-11-05T22:41:12Z

← Older revision		Revision as of 22:41, 5 November 2008
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	=Introduction=		== Introduction ==

	The use of the [[Free-Surface Green Function]] to solve the [[Standard Linear Wave Scattering Problem]]		The use of the [[Free-Surface Green Function]] to solve the [[Standard Linear Wave Scattering Problem]]
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	boundary conditions at infinite ([[Sommerfeld Radiation Condition]])		boundary conditions at infinite ([[Sommerfeld Radiation Condition]])

	= [[Standard Linear Wave Scattering Problem]] =		== [[Standard Linear Wave Scattering Problem]] ==


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	<center>		<center>
	<math>		<math>
	\frac{\partial G}{\partial z} = ~~k_{~~\~~infty}~~\phi,\,z=0.		\frac{\partial G}{\partial z} = \alpha \phi,\,z=0.
	</math>		</math>
	</center>		</center>
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	\right.		\right.
	</math></center>		</math></center>

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

2008-11-05T22:40:19Z

@@ Line 8: / Line 8: @@
 = [[Standard Linear Wave Scattering Problem]] =
 {{standard linear wave scattering equations}}
 We then use [http://en.wikipedia.org/wiki/Green's_identities Green's second identity]