Template:Standard linear wave scattering equations

The equations are the following

[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} = \mathcal{L}\phi, \, z\in\partial\Omega, }[/math]

where [math]\displaystyle{ \alpha }[/math] is the wavenumber in Infinite Depth which is given by [math]\displaystyle{ \alpha=\omega^2/g }[/math] where [math]\displaystyle{ g }[/math] is gravity. [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.