Template:Fixed body finite depth equations in two dimensions - Revision history

Meylan at 07:25, 24 August 2008

2008-08-24T07:25:31Z

@@ Line 1: / Line 1: @@
-The [[Standard Linear Wave Scattering Problem]]
+{{boundary value problem for a fixed body}}
 {{incident plane wave}}
 {{sommerfeld radiation condition two dimensions}}

Meylan at 06:27, 24 August 2008

2008-08-24T06:27:46Z

← Older revision		Revision as of 06:27, 24 August 2008
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	\frac{\partial\phi}{\partial z} = 0, \, z\in\partial\Omega.		\frac{\partial\phi}{\partial z} = 0, \, z\in\partial\Omega.
	</math></center>		</math></center>
	~~The equation is subject to some radiation conditions at infinity. We usually assume that~~
	~~there is an incident wave <math>\phi^~~{~~\mathrm~~{~~{In}}}\,</math>~~		{{incident plane wave}}
	~~is a~~ plane wave ~~travelling in the <math>x</math> direction~~
	~~<center><math>~~		{{sommerfeld radiation condition two dimensions}}
	~~\phi^{\mathrm{{In}}~~}~~({r},z)=Ae^{k_0 x}\frac{\cos k_0(z+h)}{\cos k_0 h~~}
	~~</math></center>~~
	~~where <math>A</math> is the wave amplitude and <math>k_0</math> is~~
	~~the positive imaginary solution of the [[Dispersion Relation for a Free Surface]].~~
	~~We also have to apply the [[Sommerfeld Radiation Condition]] as <math>\left\|\mathbf~~{~~r}\right\|\rightarrow~~
	~~\infty</math>.~~
	In two-dimensions ~~the condition is~~
	~~<center><math>~~
	~~\left( \frac{\partial~~}~~{\partial\|x\|~~} ~~-k_0\right)~~
	~~(\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}\|x\|\rightarrow\infty\mathrm{.}~~
	~~</math></center>~~

Meylan at 08:54, 23 August 2008

2008-08-23T08:54:26Z

← Older revision		Revision as of 08:54, 23 August 2008
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	(\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}\|x\|\rightarrow\infty\mathrm{.}		(\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}\|x\|\rightarrow\infty\mathrm{.}
	</math></center>		</math></center>
	~~where <math>k</math>~~
	~~is the wave number.~~

Meylan at 08:45, 23 August 2008

2008-08-23T08:45:31Z

← Older revision		Revision as of 08:45, 23 August 2008
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	We also have to apply the [[Sommerfeld Radiation Condition]] as <math>\left\|\mathbf{r}\right\|\rightarrow		We also have to apply the [[Sommerfeld Radiation Condition]] as <math>\left\|\mathbf{r}\right\|\rightarrow
	\infty</math>.		\infty</math>.

	In two-dimensions the condition is		In two-dimensions the condition is
	<center><math>		<center><math>
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	(\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}\|x\|\rightarrow\infty\mathrm{.}		(\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}\|x\|\rightarrow\infty\mathrm{.}
	</math></center>		</math></center>
	where ~~<math>\phi^{\mathrm{{In}}}</math> is the incident potential and~~ <math>k</math>		where <math>k</math>
	is the wave number.		is the wave number.

Meylan at 07:24, 18 August 2008

2008-08-18T07:24:46Z

← Older revision		Revision as of 07:24, 18 August 2008
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	is a plane wave travelling in the <math>x</math> direction		is a plane wave travelling in the <math>x</math> direction
	<center><math>		<center><math>
	\phi^{\mathrm{{In}}}({r},z)=Ae^{{\~~rm i}kx}\cosh k~~(z+h)\,		\phi^{\mathrm{{In}}}({r},z)=Ae^{k_0 x}\frac{\cos k_0(z+h)}{\cos k_0 h}
	</math></center>		</math></center>
	where <math>A</math> is the wave amplitude and <math>k</math> ~~is the wavenumber which~~ is		where <math>A</math> is the wave amplitude and <math>k_0</math> is
	the positive ~~real~~ solution of the [[Dispersion Relation for a Free Surface]].		the positive imaginary solution of the [[Dispersion Relation for a Free Surface]].
	We also have to apply the [[Sommerfeld Radiation Condition]] as <math>\left\|\mathbf{r}\right\|\rightarrow		We also have to apply the [[Sommerfeld Radiation Condition]] as <math>\left\|\mathbf{r}\right\|\rightarrow
	\infty</math>.		\infty</math>.
Line 26:		Line 26:
	In two-dimensions the condition is		In two-dimensions the condition is
	<center><math>		<center><math>
	\left( \frac{\partial}{\partial\|x\|}-~~{i}k~~\right)		\left( \frac{\partial}{\partial\|x\|} -k_0\right)
	(\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}\|x\|\rightarrow\infty\mathrm{.}		(\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}\|x\|\rightarrow\infty\mathrm{.}
	</math></center>		</math></center>
	where <math>\phi^{\mathrm{{In}}}</math> is the incident potential and <math>k</math>		where <math>\phi^{\mathrm{{In}}}</math> is the incident potential and <math>k</math>
	is the wave number.		is the wave number.

Meylan at 02:59, 18 August 2008

2008-08-18T02:59:05Z

← Older revision		Revision as of 02:59, 18 August 2008
Line 13:		Line 13:
	\frac{\partial\phi}{\partial z} = 0, \, z\in\partial\Omega.		\frac{\partial\phi}{\partial z} = 0, \, z\in\partial\Omega.
	</math></center>		</math></center>
			The equation is subject to some radiation conditions at infinity. We usually assume that
			there is an incident wave <math>\phi^{\mathrm{{In}}}\,</math>
			is a plane wave travelling in the <math>x</math> direction
			<center><math>
			\phi^{\mathrm{{In}}}({r},z)=Ae^{{\rm i}kx}\cosh k(z+h)\,
			</math></center>
			where <math>A</math> is the wave amplitude and <math>k</math> is the wavenumber which is
			the positive real solution of the [[Dispersion Relation for a Free Surface]].
			We also have to apply the [[Sommerfeld Radiation Condition]] as <math>\left\|\mathbf{r}\right\|\rightarrow
			\infty</math>.

			In two-dimensions the condition is
			<center><math>
			\left( \frac{\partial}{\partial\|x\|}-{i}k\right)
			(\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}\|x\|\rightarrow\infty\mathrm{.}
			</math></center>
			where <math>\phi^{\mathrm{{In}}}</math> is the incident potential and <math>k</math>
			is the wave number.

Meylan at 02:56, 18 August 2008

2008-08-18T02:56:15Z

New page

The [[Standard Linear Wave Scattering Problem]]
in [[Finite Depth]] for a fixed body in two dimensions is
<center><math>
\nabla^{2}\phi=0, \, -h<z<0,\,\,\,\mathbf{x}\notin \Omega
</math></center>
<center><math>
\frac{\partial\phi}{\partial z}=0, \, z=-h,
</math></center>
<center><math>
\frac{\partial\phi}{\partial z} = \alpha \phi,\,z=0,\,\,\mathbf{x}\notin\Omega,
</math></center>
<center><math>
\frac{\partial\phi}{\partial z} = 0, \, z\in\partial\Omega.
</math></center>