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| | The [[Standard Linear Wave Scattering Problem]] | | The [[Standard Linear Wave Scattering Problem]] |
| | in [[Finite Depth]] for a fixed body is | | in [[Finite Depth]] for a fixed body is |
| − | <center><math>
| + | {{standard linear wave scattering equations without body condition}} |
| − | \nabla^{2}\phi=0, \, -h<z<0,\,\,\,\mathbf{x}\notin \Omega
| + | {{frequency domain equations for a rigid body}} |
| − | </math></center>
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| − | <center><math>
| |
| − | \frac{\partial\phi}{\partial z}=0, \, z=-h,
| |
| − | </math></center>
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| − | <center><math>
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| − | \frac{\partial\phi}{\partial z} = \alpha \phi,\,z=0,\,\,\mathbf{x}\notin\Omega,
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| − | </math></center>
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| − | <center><math>
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| − | \frac{\partial\phi}{\partial n} = 0, \, \mathbf{x}\in\partial\Omega.
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| − | </math></center>
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| − | where <math>n</math> is the outward (from the fluid) normal.
| |
Latest revision as of 01:42, 21 August 2009
The Standard Linear Wave Scattering Problem
in Finite Depth for a fixed body is
[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])
The body boundary condition for a rigid body is just
[math]\displaystyle{
\partial_{n}\phi=0,\ \ \mathbf{x}\in\partial\Omega_{\mathrm{B}},
}[/math]
