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| | The equations are the following | | The equations are the following |
| − | <center><math>
| + | {{standard linear wave scattering equations without body condition}} |
| − | \nabla^{2}\phi=0, \, -h<z<0,\,\,\,\mathbf{x}\notin \Omega
| + | {{general body boundary condition}} |
| − | </math></center>
| |
| − | <center><math>
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| − | \frac{\partial\phi}{\partial z}=0, \, z=-h,
| |
| − | </math></center>
| |
| − | <center><math>
| |
| − | \frac{\partial\phi}{\partial z} = \alpha \phi,\,z=0,\,\,\mathbf{x}\notin\Omega,
| |
| − | </math></center>
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| − | <center><math>
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| − | \frac{\partial\phi}{\partial z} = L\phi, \, z\in\partial\Omega,
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| − | </math></center>
| |
| − | | |
| − | where <math>\alpha</math> is the wavenumber in [[Infinite Depth]] which is given by
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| − | <math>\alpha=\omega^2/g</math> where <math>g</math> is gravity. <math>L</math> is a linear
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| − | operator which relates the normal and potential on the body surface through the physics
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| − | of the body.
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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.
