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]