Template:Standard linear wave scattering equations without body condition

[math]\displaystyle{ \Delta\phi=0, \, -h\lt z\lt 0,\,\,\,\mathbf{x} \in \Omega }[/math] [math]\displaystyle{ \partial_z\phi = 0, \, z=-h, }[/math] [math]\displaystyle{ \partial_z \phi = \alpha \phi,\,z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, }[/math]

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

[math]\displaystyle{ \partial_z \phi =\mathrm{i} \omega \zeta,\,z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, }[/math] [math]\displaystyle{ \mathrm{i} \omega \phi = -g\zeta,\,z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, }[/math]

where [math]\displaystyle{ \alpha = \omega^2/g }[/math])