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)
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 begin with the Standard Linear Wave Scattering Problem
[math]\displaystyle{
\nabla^{2}\phi=0, \, -\infty\lt z\lt 0,\,\,\,\mathbf{x}\notin \Omega
}[/math]
[math]\displaystyle{
\frac{\partial\phi}{\partial z}=0, \, z=h,
}[/math]
[math]\displaystyle{
\frac{\partial\phi}{\partial z} = k_{\infty}\phi,\,z=0,\,\,\mathbf{x}\notin\Omega,
}[/math]
[math]\displaystyle{
\frac{\partial\phi}{\partial z} = L\phi, \, z\in\partial\Omega,
}[/math]
We then use Green's second identity
If φ and ψ are both twice continuously differentiable on U, then
[math]\displaystyle{ \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]
If we then substitiute 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}), \, -\infty\lt z\lt 0
}[/math]
[math]\displaystyle{
\frac{\partial G}{\partial z}=0, \, z=-h,
}[/math]
[math]\displaystyle{
\frac{\partial G}{\partial z} = k_{\infty}\phi,\,z=0.
}[/math]
for ψ we obtain
[math]\displaystyle{
\phi^{in} +
\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 U \cup \partial U, \\
\phi(\mathbf{x})/2,\,\,\,\mathbf{x} \in \partial U, \\
\phi(\mathbf{x}),\,\,\,\mathbf{x} \in U,
\end{matrix}
\right.
}[/math]
