Difference between revisions of "Green Function Solution Method"

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=Introduction=
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{{complete pages}}
 +
 
 +
== 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]] =
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== [[Standard Linear Wave Scattering Problem]] ==
 +
 
  
We begin with the [[Standard Linear Wave Scattering Problem]]  
+
We begin with the [[Standard Linear Wave Scattering Problem]].
<center><math>
+
{{standard linear wave scattering equations}}
\nabla^{2}\phi=0, \, -\infty<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} = k_{\infty}\phi,\,z=0,\,\,\mathbf{x}\notin\Omega,
 
</math></center>
 
<center><math>
 
\frac{\partial\phi}{\partial z} = L\phi, \, z\in\partial\Omega,
 
</math></center>
 
  
 
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 &phi; and &psi; are both twice continuously differentiable on ''U'', then
+
If &phi; and &psi; 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 substitiute the [[Free-Surface Green Function]] which satisfies the following equations
+
If we then substitute the [[Free-Surface Green Function]] which satisfies the following equations (plus the
 +
[[Sommerfeld Radiation Condition]] far from the body)
 
<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>
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<center>
 
<center>
 
<math>
 
<math>
  \frac{\partial G}{\partial z} = k_{\infty}\phi,\,z=0.
+
  \frac{\partial G}{\partial z} = \alpha G,\,z=0.
 
</math>
 
</math>
 
</center>
 
</center>
 
for &psi; we obtain  
 
for &psi; we obtain  
 +
<center><math>
 +
\phi^\mathrm{I}  +
 +
\int_{\partial \Omega }\left(
 +
G_{n}\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi \left( \mathbf{x}
 +
^{\prime }\right) -G\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi
 +
_{n}\left( \mathbf{x}^{\prime }\right) \right) d\mathbf{x}^{\prime }
 +
  =
 +
\left(
 +
\begin{matrix}
 +
0, \,\,\,x\notin \Omega \cup \partial \Omega, \\
 +
\phi(\mathbf{x})/2,\,\,\,\mathbf{x} \in \partial \Omega, \\
 +
\phi(\mathbf{x}),\,\,\,\mathbf{x} \in \Omega,
 +
\end{matrix}
 +
\right.
 +
</math></center>
 +
 
[[Category:Linear Water-Wave Theory]]
 
[[Category:Linear Water-Wave Theory]]

Latest revision as of 19:34, 8 February 2010


Introduction

The use of the Free-Surface Green Function to solve the Standard Linear Wave Scattering Problem has proved one of the most powerful methods, primarily because of its very general nature so that it can deal with complicated boundary conditions. It also solves explicity for the boundary conditions at infinite (Sommerfeld Radiation Condition)

Standard Linear Wave Scattering Problem

We begin with the Standard Linear Wave Scattering Problem. The equations are the following

[math]\displaystyle{ \begin{align} \Delta\phi &=0, &-h\lt z\lt 0,\,\,\mathbf{x} \in \Omega \\ \partial_z\phi &= 0, &z=-h, \\ \partial_z \phi &= \alpha \phi, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \end{align} }[/math]


(note that the last expression can be obtained from combining the expressions:

[math]\displaystyle{ \begin{align} \partial_z \phi &= -\mathrm{i} \omega \zeta, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \\ \mathrm{i} \omega \phi &= g\zeta, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \end{align} }[/math]

where [math]\displaystyle{ \alpha = \omega^2/g \, }[/math])

[math]\displaystyle{ \partial_n\phi = \mathcal{L}\phi, \quad \mathbf{x}\in\partial\Omega_B, }[/math]

where [math]\displaystyle{ \mathcal{L} }[/math] is a linear operator which relates the normal and potential on the body surface through the physics of the body.

We then use Green's second identity If φ and ψ are both twice continuously differentiable on [math]\displaystyle{ \Omega }[/math], then

[math]\displaystyle{ \int_\Omega \left( \psi \nabla^2 \varphi - \varphi \nabla^2 \psi\right)\, dV = \oint_{\partial \Omega} \left( \psi {\partial \varphi \over \partial n} - \varphi {\partial \psi \over \partial n}\right)\, dS }[/math]

If we then substitute the Free-Surface Green Function which satisfies the following equations (plus the Sommerfeld Radiation Condition far from the body)

[math]\displaystyle{ \nabla_{\mathbf{x}}^{2}G(\mathbf{x},\mathbf{\xi})=\delta(\mathbf{x}-\mathbf{\xi}), \, -h\lt z\lt 0 }[/math]

[math]\displaystyle{ \frac{\partial G}{\partial z}=0, \, z=-h, }[/math]

[math]\displaystyle{ \frac{\partial G}{\partial z} = \alpha G,\,z=0. }[/math]

for ψ we obtain

[math]\displaystyle{ \phi^\mathrm{I} + \int_{\partial \Omega }\left( G_{n}\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi \left( \mathbf{x} ^{\prime }\right) -G\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi _{n}\left( \mathbf{x}^{\prime }\right) \right) d\mathbf{x}^{\prime } = \left( \begin{matrix} 0, \,\,\,x\notin \Omega \cup \partial \Omega, \\ \phi(\mathbf{x})/2,\,\,\,\mathbf{x} \in \partial \Omega, \\ \phi(\mathbf{x}),\,\,\,\mathbf{x} \in \Omega, \end{matrix} \right. }[/math]