# Template:Fixed body finite depth equations in two dimensions

The Standard Linear Wave Scattering Problem in Finite Depth for a fixed body is

\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} }

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

\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} }

where $\displaystyle{ \alpha = \omega^2/g \, }$) The body boundary condition for a rigid body is just

$\displaystyle{ \partial_{n}\phi=0,\ \ \mathbf{x}\in\partial\Omega_{\mathrm{B}}, }$

The equation is subject to some radiation conditions at infinity. We assume the following. $\displaystyle{ \phi^{\mathrm{I}}\, }$ is a plane wave travelling in the $\displaystyle{ x }$ direction,

$\displaystyle{ \phi^{\mathrm{I}}(x,z)=A \phi_0(z) e^{\mathrm{i} k x} \, }$

where $\displaystyle{ A }$ is the wave amplitude (in potential) $\displaystyle{ \mathrm{i} k }$ is the positive imaginary solution of the Dispersion Relation for a Free Surface (note we are assuming that the time dependence is of the form $\displaystyle{ \exp(-\mathrm{i}\omega t) }$) and

$\displaystyle{ \phi_0(z) =\frac{\cosh k(z+h)}{\cosh k h} }$

In two-dimensions the Sommerfeld Radiation Condition is

$\displaystyle{ \left( \frac{\partial}{\partial|x|} - \mathrm{i} k \right) (\phi-\phi^{\mathrm{{I}}})=0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.} }$

where $\displaystyle{ \phi^{\mathrm{{I}}} }$ is the incident potential.