Removing the Depth Dependence - Revision history

Meylan at 19:15, 8 February 2010

2010-02-08T19:15:48Z

← Older revision		Revision as of 19:15, 8 February 2010
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			{{complete pages}}

	We are considering the [[Frequency Domain Problem]] for linear wave waves.		We are considering the [[Frequency Domain Problem]] for linear wave waves.

Meylan at 08:57, 24 August 2008

2008-08-24T08:57:44Z

← Older revision		Revision as of 08:57, 24 August 2008
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	We are considering the [[Frequency Domain Problem]] for linear wave waves.		We are considering the [[Frequency Domain Problem]] for linear wave waves.
	{{fixed body ~~finite depth equations in~~ two dimensions}}
			{{standard linear problem notation}}

			{{boundary value problem for a fixed body}}

			{{incident plane wave}}

			{{sommerfeld radiation condition two dimensions}}

	{{removing the depth dependence}}		{{removing the depth dependence}}

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

Meylan at 08:55, 24 August 2008

2008-08-24T08:55:09Z

@@ Line 2: / Line 2: @@
 {{fixed body finite depth equations in two dimensions}}
-If we have a problem in which all the scatterers are of constant cross sections so
+{{removing the depth dependence}}
-that
 [[Category:Linear Water-Wave Theory]]

Meylan at 09:07, 23 August 2008

2008-08-23T09:07:23Z

← Older revision		Revision as of 09:07, 23 August 2008
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	because <math>k_0</math> is the pure imaginary, and it is more normal to write		because <math>k_0</math> is the pure imaginary, and it is more normal to write
	<center>		<center>
	<math>\nabla^2 \Phi - k*2 \Phi = 0 </math>		<math>\nabla^2 \Phi - k^2 \Phi = 0 </math>
	</center>		</center>
	where <math>k=-ik_0.</math>		where <math>k=-ik_0.</math>
	[[Category:Linear Water-Wave Theory]]		[[Category:Linear Water-Wave Theory]]

Meylan at 09:02, 23 August 2008

2008-08-23T09:02:45Z

← Older revision		Revision as of 09:02, 23 August 2008
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	Since <math>\phi</math> satisfies [[Laplace's Equation]], then <math>\Phi</math> satisfies [[Helmholtz's Equation]]		Since <math>\phi</math> satisfies [[Laplace's Equation]], then <math>\Phi</math> satisfies [[Helmholtz's Equation]]
	<center>		<center>
	<math>\nabla^2 \~~phi~~ + k_0^2 \~~phi~~ = 0 </math>		<math>\nabla^2 \Phi + k_0^2 \Phi = 0 </math>
	</center>		</center>
	in the region not occupied by the scatterers. Not that this is not the standard why to write [[Helmholtz's Equation]]		in the region not occupied by the scatterers. Not that this is not the standard why to write [[Helmholtz's Equation]]
	because <math>k_0</math> is the pure imaginary, and it is more normal to write		because <math>k_0</math> is the pure imaginary, and it is more normal to write
	<center>		<center>
	<math>\nabla^2 \~~phi~~ - k*2 \~~phi~~ = 0 </math>		<math>\nabla^2 \Phi - k*2 \Phi = 0 </math>
	</center>		</center>
	where <math>k=-ik_0.</math>		where <math>k=-ik_0.</math>
	[[Category:Linear Water-Wave Theory]]		[[Category:Linear Water-Wave Theory]]

Meylan at 09:01, 23 August 2008

2008-08-23T09:01:37Z

← Older revision		Revision as of 09:01, 23 August 2008
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	where <math>\partial\hat{\Omega} </math> is a function only of <math>x,y</math>		where <math>\partial\hat{\Omega} </math> is a function only of <math>x,y</math>
	i.e. the boundary of the scattering bodies is uniform with respect to depth.		i.e. the boundary of the scattering bodies is uniform with respect to depth.
	~~Then we~~ can remove the depth dependence ~~by assuming~~		We can remove the depth dependence [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables]
	that the dependence on depth is given by		and obtain that the dependence on depth is given by
	<center>		<center>
	<math>		<math>
	\~~Phi~~(x,y,z) = \frac{\~~cosh~~ \big( k (z+h) \big)}{\~~cosh~~(k h)} \~~phi~~(x,y)		\phi(x,y,z) = \frac{\cos \big( k_0 (z+h) \big)}{\cos(k_0 h)} \Phi(x,y)
	</math>		</math>
	</center>		</center>
	~~where <math>k</math> is the positive root of the [[Dispersion Relation for a Free Surface]].~~		Since <math>\phi</math> satisfies [[Laplace's Equation]], then <math>\Phi</math> satisfies [[Helmholtz's Equation]]
	Since <math>\~~Phi~~</math> satisfies [[Laplace's Equation]], then <math>\~~phi~~</math> satisfies [[Helmholtz's Equation]]
	<center>		<center>
	<math>\nabla^2 \phi ~~- k~~^2 \phi = 0 </math>		<math>\nabla^2 \phi + k_0^2 \phi = 0 </math>
	</center>		</center>
	in the region not occupied by the scatterers.		in the region not occupied by the scatterers. Not that this is not the standard why to write [[Helmholtz's Equation]]
			because <math>k_0</math> is the pure imaginary, and it is more normal to write
			<center>
			<math>\nabla^2 \phi - k*2 \phi = 0 </math>
			</center>
			where <math>k=-ik_0.</math>
	[[Category:Linear Water-Wave Theory]]		[[Category:Linear Water-Wave Theory]]

Meylan at 08:54, 23 August 2008

2008-08-23T08:54:30Z

← Older revision		Revision as of 08:54, 23 August 2008
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	We are considering the [[Frequency Domain Problem]] for linear wave waves.		We are considering the [[Frequency Domain Problem]] for linear wave waves.
	If we have a problem in which the ~~water depth is~~ of constant ~~depth~~ <math>z=-h </math> ~~(we are assuming~~		{{fixed body finite depth equations in two dimensions}}
	~~the free surface~~ is at <math>~~z=0~~</math>~~) and all~~ the ~~scatterers are~~ of ~~constant cross sections and~~
	~~extend throughout~~ the depth ~~then~~ we can remove the depth dependence by assuming		If we have a problem in which all the scatterers are of constant cross sections so
			that
			<center>
			<math>\partial\Omega = \partial\hat{\Omega} \times z\in[-h,0]
			</math>
			</center>
			where <math>\partial\hat{\Omega} </math> is a function only of <math>x,y</math>
			i.e. the boundary of the scattering bodies is uniform with respect to depth.
			Then we can remove the depth dependence by assuming
	that the dependence on depth is given by		that the dependence on depth is given by
	<center>		<center>

Meylan at 08:38, 23 August 2008

2008-08-23T08:38:57Z

← Older revision		Revision as of 08:38, 23 August 2008
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	<center>		<center>
	<math>		<math>
	\Phi(x,y,z) = \cosh \big( k (z+h) \big) \phi(x,y)		\Phi(x,y,z) = \frac{\cosh \big( k (z+h) \big)}{\cosh(k h)} \phi(x,y)
	</math>		</math>
	</center>		</center>

	where <math>k</math> is the positive root of the [[Dispersion Relation for a Free Surface]].		where <math>k</math> is the positive root of the [[Dispersion Relation for a Free Surface]].
	Since <math>\Phi</math> satisfies [[Laplace's Equation]], then <math>\phi</math> satisfies [[Helmholtz's Equation]]		Since <math>\Phi</math> satisfies [[Laplace's Equation]], then <math>\phi</math> satisfies [[Helmholtz's Equation]]

Meylan at 14:17, 22 August 2008

2008-08-22T14:17:31Z

← Older revision		Revision as of 14:17, 22 August 2008
Line 4:		Line 4:
	extend throughout the depth then we can remove the depth dependence by assuming		extend throughout the depth then we can remove the depth dependence by assuming
	that the dependence on depth is given by		that the dependence on depth is given by
			<center>
	<math>		<math>
	\Phi(x,y,z) = \cosh \big( k (z+h) \big) \phi(x,y)		\Phi(x,y,z) = \cosh \big( k (z+h) \big) \phi(x,y)
	</math>		</math>
			</center>

	where <math>k</math> is the positive root of the [[Dispersion Relation for a Free Surface]].		where <math>k</math> is the positive root of the [[Dispersion Relation for a Free Surface]].
	Since <math>\Phi</math> satisfies [[Laplace's Equation]], then <math>\phi</math> satisfies [[Helmholtz's Equation]]		Since <math>\Phi</math> satisfies [[Laplace's Equation]], then <math>\phi</math> satisfies [[Helmholtz's Equation]]
			<center>
	<math>\nabla^2 \phi - k^2 \phi = 0 </math>		<math>\nabla^2 \phi - k^2 \phi = 0 </math>
			</center>
	in the region not occupied by the scatterers.		in the region not occupied by the scatterers.

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

Adi Kurniawan at 05:55, 22 August 2008

2008-08-22T05:55:15Z

← Older revision		Revision as of 05:55, 22 August 2008
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	We are considering the [[Frequency Domain Problem]] for linear wave waves.		We are considering the [[Frequency Domain Problem]] for linear wave waves.
	If we have a problem in which the water depth is of constant depth <math>z=-d </math> (we are assuming		If we have a problem in which the water depth is of constant depth <math>z=-h </math> (we are assuming
	the free surface is at <math>z=0</math>) and all the scatterers are of constant cross sections and		the free surface is at <math>z=0</math>) and all the scatterers are of constant cross sections and
	extend throughout the depth then we can remove the depth dependence by assuming		extend throughout the depth then we can remove the depth dependence by assuming
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	<math>		<math>
	\Phi(x,y,z) = \cosh \big( k (z+d) \big) \phi(x,y)		\Phi(x,y,z) = \cosh \big( k (z+h) \big) \phi(x,y)
	</math>		</math>