Template:Fixed body finite depth equations in two dimensions

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

[math]\displaystyle{ \nabla^{2}\phi=0, \, -h\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} = \alpha \phi,\,z=0,\,\,\mathbf{x}\notin\Omega, }[/math] [math]\displaystyle{ \frac{\partial\phi}{\partial z} = 0, \, z\in\partial\Omega. }[/math]

The equation is subject to some radiation conditions at infinity. We usually assume that there is an incident wave [math]\displaystyle{ \phi^{\mathrm{{In}}}\, }[/math] is a plane wave travelling in the [math]\displaystyle{ x }[/math] direction

[math]\displaystyle{ \phi^{\mathrm{{In}}}({r},z)=Ae^{k_0 x}\frac{\cos k_0(z+h)}{\cos k_0 h} }[/math]

where [math]\displaystyle{ A }[/math] is the wave amplitude and [math]\displaystyle{ k_0 }[/math] is the positive imaginary solution of the Dispersion Relation for a Free Surface. We also have to apply the Sommerfeld Radiation Condition as [math]\displaystyle{ \left|\mathbf{r}\right|\rightarrow \infty }[/math]. In two-dimensions the condition is

[math]\displaystyle{ \left( \frac{\partial}{\partial|x|} -k_0\right) (\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.} }[/math]

where [math]\displaystyle{ k }[/math] is the wave number.