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| − | The [[Standard Linear Wave Scattering Problem]]
| + | {{boundary value problem for a fixed body}} |
| − | in [[Finite Depth]] for a fixed body in two dimensions is
| + | |
| − | <center><math>
| + | {{incident plane wave}} |
| − | \nabla^{2}\phi=0, \, -h<z<0,\,\,\,\mathbf{x}\notin \Omega
| + | |
| − | </math></center>
| + | {{sommerfeld radiation condition two dimensions}} |
| − | <center><math>
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| − | \frac{\partial\phi}{\partial z}=0, \, z=-h,
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| − | </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 z} = 0, \, z\in\partial\Omega.
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| − | </math></center>
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| − | The equation is subject to some radiation conditions at infinity. We usually assume that
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| − | there is an incident wave <math>\phi^{\mathrm{{In}}}\,</math>
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| − | is a plane wave travelling in the <math>x</math> direction
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| − | <center><math>
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| − | \phi^{\mathrm{{In}}}({r},z)=Ae^{k_0 x}\frac{\cos k_0(z+h)}{\cos k_0 h}
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| − | </math></center>
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| − | where <math>A</math> is the wave amplitude and <math>k_0</math> is
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| − | the positive imaginary solution of the [[Dispersion Relation for a Free Surface]].
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| − | We also have to apply the [[Sommerfeld Radiation Condition]] as <math>\left|\mathbf{r}\right|\rightarrow
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| − | \infty</math>.
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| − | In two-dimensions the condition is
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| − | <center><math>
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| − | \left( \frac{\partial}{\partial|x|} -k_0\right)
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| − | (\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.}
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| − | </math></center>
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| − | where <math>k</math>
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| − | is the wave number.
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Latest revision as of 07:25, 24 August 2008
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]
The equation is subject to some radiation conditions at infinity. We assume the following.
[math]\displaystyle{ \phi^{\mathrm{I}}\, }[/math]
is a plane wave travelling in the [math]\displaystyle{ x }[/math] direction,
[math]\displaystyle{
\phi^{\mathrm{I}}(x,z)=A \phi_0(z) e^{\mathrm{i} k x} \,
}[/math]
where [math]\displaystyle{ A }[/math] is the wave amplitude (in potential) [math]\displaystyle{ \mathrm{i} k }[/math] is
the positive imaginary solution of the Dispersion Relation for a Free Surface
(note we are assuming that the time dependence is of the form [math]\displaystyle{ \exp(-\mathrm{i}\omega t) }[/math])
and
[math]\displaystyle{
\phi_0(z) =\frac{\cosh k(z+h)}{\cosh k h}
}[/math]
In two-dimensions the Sommerfeld Radiation Condition is
[math]\displaystyle{
\left( \frac{\partial}{\partial|x|} - \mathrm{i} k \right)
(\phi-\phi^{\mathrm{{I}}})=0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.}
}[/math]
where [math]\displaystyle{ \phi^{\mathrm{{I}}} }[/math] is the incident potential.
