Template:Standard linear wave scattering equations

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The equations are the following

[math] \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} [/math]

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

[math] \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} [/math]

where [math]\alpha = \omega^2/g \,[/math])

[math] \partial_n\phi = \mathcal{L}\phi, \quad \mathbf{x}\in\partial\Omega_B, [/math]

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