View source for Green Function Solution Method

== 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]].
{{standard linear wave scattering equations}}

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
<center><math> \int_U \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 
</math></center>
If we then substitute the [[Free-Surface Green Function]] which satisfies the following equations (plus the 
[[Sommerfeld Radiation Condition]] far from the body)
<center>
<math>
\nabla_{\mathbf{x}}^{2}G(\mathbf{x},\mathbf{\xi})=\delta(\mathbf{x}-\mathbf{\xi}), \, -h<z<0
</math>
</center>
<center>
<math>
\frac{\partial G}{\partial z}=0, \, z=-h,
</math>
</center>
<center>
<math>
 \frac{\partial G}{\partial z} = \alpha G,\,z=0.
</math>
</center>
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]]