The Standard Linear Wave Scattering Problem
in Finite Depth for a fixed body is
[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])
The body boundary condition for a rigid body is just
[math]\displaystyle{
\partial_{n}\phi=0,\ \ \mathbf{x}\in\partial\Omega_{\mathrm{B}},
}[/math]
