Standard Linear Wave Scattering Problem - Revision history

Meylan at 19:14, 8 February 2010

2010-02-08T19:14:45Z

← Older revision		Revision as of 19:14, 8 February 2010
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			{{complete pages}}

	{{standard linear problem notation}}		{{standard linear problem notation}}
	[[Variable Bottom Topography]]		[[Variable Bottom Topography]]

Meylan at 07:31, 24 August 2008

2008-08-24T07:31:41Z

← Older revision		Revision as of 07:31, 24 August 2008
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	{{standard linear problem notation}}		{{standard linear problem notation}}
			[[Variable Bottom Topography]]
			can also easily be included.

	{{standard linear wave scattering equations}}		{{standard linear wave scattering equations}}

Meylan at 07:30, 24 August 2008

2008-08-24T07:30:02Z

← Older revision		Revision as of 07:30, 24 August 2008
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	~~The~~ standard linear ~~wave scattering~~ problem ~~is based on assuming irrotational and inviscid fluid motion and the wave~~		{{standard linear problem notation}}
	~~sufficiently small amplitude so that we can linearise all the~~
	~~equations. We also assume that [[Frequency Domain Problem]] with frequency <math>\omega</math>~~
	~~The water motion is represented by a velocity potential which is~~
	~~denoted by <math>\phi</math>. The coordinate system is the standard Cartesian coordinate system~~
	~~with the <math>z</math> axis pointing vertically up. The water surface is at~~
	~~<math>z=0</math> and the region of interest is~~
	~~<math>-h<z<0</math>. There is a body which occupies the region <math>\Omega</math>~~
	~~and we denoted the wetted surface of the body by <math>\partial\Omega</math>~~
	~~We denote <math>\mathbf{r~~}~~=(x,y)</math> as the horizontal coordinate in two or three dimensions~~
	~~respectively and the cartesian system we denote by <math>\mathbf{x~~}~~</math>.~~
	~~We assume that the bottom surface is of constant depth but [[Variable Bottom Topography]]~~
	~~can easily be included.~~

	{{standard linear wave scattering equations}}		{{standard linear wave scattering equations}}

Meylan at 07:20, 24 August 2008

2008-08-24T07:20:17Z

← Older revision		Revision as of 07:20, 24 August 2008
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	where the operator is <math>L=0</math> but more complicated conditions are possible.		where the operator is <math>L=0</math> but more complicated conditions are possible.

	~~The equation is subject to some radiation conditions at infinity. We usually assume that~~		{{incident plane wave}}
	~~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>.~~

	{{sommerfeld radiation condition two dimensions}}		{{sommerfeld radiation condition two dimensions}}

Meylan at 06:22, 24 August 2008

2008-08-24T06:22:37Z

← Older revision		Revision as of 06:22, 24 August 2008
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	{{sommerfeld radiation condition three dimensions}}		{{sommerfeld radiation condition three dimensions}}
	~~In three-dimensions the [[Sommerfeld Radiation Condition]] is~~
	~~<center><math>~~
	~~\sqrt{\|\mathbf{r}\|}\left( \frac{\partial}{\partial\|\mathbf{r}\|}-{i}k\right)~~
	~~(\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}\|\mathbf{r}\|\rightarrow\infty\mathrm{.}~~
	~~</math></center>~~

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

Meylan at 06:21, 24 August 2008

2008-08-24T06:21:40Z

← Older revision		Revision as of 06:21, 24 August 2008
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	\infty</math>.		\infty</math>.

	In two-dimensions ~~the condition is~~		{{sommerfeld radiation condition two dimensions}}
	~~<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.~~

	In three-dimensions the ~~condition~~ is		{{sommerfeld radiation condition three dimensions}}
			In three-dimensions the [[Sommerfeld Radiation Condition]] is
	<center><math>		<center><math>
	\sqrt{\|\mathbf{r}\|}\left( \frac{\partial}{\partial\|\mathbf{r}\|}-{i}k\right)		\sqrt{\|\mathbf{r}\|}\left( \frac{\partial}{\partial\|\mathbf{r}\|}-{i}k\right)

Meylan at 21:44, 18 August 2008

2008-08-18T21:44:05Z

← Older revision		Revision as of 21:44, 18 August 2008
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	can easily be included.		can easily be included.

	~~The equations are the following~~		{{standard linear wave scattering equations}}
	~~<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~~} ~~= L\phi, \, z\in\partial\Omega,~~
	~~</math></center>~~

	~~where <math>\alpha</math> is the wavenumber in [[Infinite Depth]] which is given by~~		The simplest case is for a fixed body
	~~<math>\alpha=\omega^2/g</math> where <math>g</math> is gravity. <math>L</math> is a linear~~
	~~operator which relates the normal and potential on the body surface through the physics~~
	~~of the body.~~ The simplest case is for a fixed body
	where the operator is <math>L=0</math> but more complicated conditions are possible.		where the operator is <math>L=0</math> but more complicated conditions are possible.

Meylan at 22:45, 1 April 2008

2008-04-01T22:45:37Z

← Older revision		Revision as of 22:45, 1 April 2008
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	The equations are the following		The equations are the following
	<center><math>		<center><math>
	\nabla^{2}\phi=0, \, -~~\infty~~<z<0,\,\,\,\mathbf{x}\notin \Omega		\nabla^{2}\phi=0, \, -h<z<0,\,\,\,\mathbf{x}\notin \Omega
	</math></center>		</math></center>
	<center><math>		<center><math>
	\frac{\partial\phi}{\partial z}=0, \, z=h,		\frac{\partial\phi}{\partial z}=0, \, z=-h,
	</math></center>		</math></center>
	<center><math>		<center><math>

Meylan at 21:51, 6 March 2008

2008-03-06T21:51:28Z

← Older revision		Revision as of 21:51, 6 March 2008
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	</math></center>		</math></center>
	<center><math>		<center><math>
	\frac{\partial\phi}{\partial z} = \~~apha~~ \phi,\,z=0,\,\,\mathbf{x}\notin\Omega,		\frac{\partial\phi}{\partial z} = \alpha \phi,\,z=0,\,\,\mathbf{x}\notin\Omega,
	</math></center>		</math></center>
	<center><math>		<center><math>
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	</math></center>		</math></center>

	where <math>\~~aklpha~~</math> is the wavenumber in [[Infinite Depth]] which is given by		where <math>\alpha</math> is the wavenumber in [[Infinite Depth]] which is given by
	<math>\~~aplha~~=\omega^2/g</math> where <math>g</math> is gravity. <math>L</math> is a linear		<math>\alpha=\omega^2/g</math> where <math>g</math> is gravity. <math>L</math> is a linear
	operator which relates the normal and potential on the body surface through the physics		operator which relates the normal and potential on the body surface through the physics
	of the body. The simplest case is for a fixed body		of the body. The simplest case is for a fixed body

Meylan at 21:51, 6 March 2008

2008-03-06T21:51:02Z

← Older revision		Revision as of 21:51, 6 March 2008
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	</math></center>		</math></center>
	<center><math>		<center><math>
	\frac{\partial\phi}{\partial z} = ~~k_{~~\~~infty}~~\phi,\,z=0,\,\,\mathbf{x}\notin\Omega,		\frac{\partial\phi}{\partial z} = \apha \phi,\,z=0,\,\,\mathbf{x}\notin\Omega,
	</math></center>		</math></center>
	<center><math>		<center><math>
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	</math></center>		</math></center>

	where <math>~~k_{~~\~~infty}~~</math> is the wavenumber in [[Infinite Depth]] which is given by		where <math>\aklpha</math> is the wavenumber in [[Infinite Depth]] which is given by
	<math>~~k_{~~\~~infty}~~=\omega^2/g</math> where <math>g</math> is gravity. <math>L</math> is a linear		<math>\aplha=\omega^2/g</math> where <math>g</math> is gravity. <math>L</math> is a linear
	operator which relates the normal and potential on the body surface through the physics		operator which relates the normal and potential on the body surface through the physics
	of the body. The simplest case is for a fixed body		of the body. The simplest case is for a fixed body