|
|
| Line 1: |
Line 1: |
| | The equations are the following | | The equations are the following |
| | {{standard linear wave scattering equations without body condition}} | | {{standard linear wave scattering equations without body condition}} |
| − | <center><math>
| + | {{general body boundary condition}} |
| − | \partial_n\phi = \mathcal{L}\phi, \, z\in\partial\Omega,
| |
| − | </math></center>
| |
| − | where <math>\alpha</math> is the wavenumber in [[Infinite Depth]] which is given by
| |
| − | <math>\alpha=\omega^2/g</math> where <math>g</math> is gravity. <math>\mathcal{L}</math> is a linear
| |
| − | operator which relates the normal and potential on the body surface through the physics
| |
| − | of the body.
| |
Latest revision as of 04:20, 20 August 2009
The equations are the following
[math]\displaystyle{
\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]\displaystyle{
\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]\displaystyle{ \alpha = \omega^2/g \, }[/math])
[math]\displaystyle{
\partial_n\phi = \mathcal{L}\phi, \quad \mathbf{x}\in\partial\Omega_B,
}[/math]
where [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.
