# Template:Standard linear wave scattering equations

The equations are the following

\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:

\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 $\alpha = \omega^2/g \,$)

$\partial_n\phi = \mathcal{L}\phi, \quad \mathbf{x}\in\partial\Omega_B,$

where $\mathcal{L}$ is a linear operator which relates the normal and potential on the body surface through the physics of the body.