|
|
| Line 13: |
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| \frac{\partial\phi}{\partial z} = 0, \, z\in\partial\Omega. | | \frac{\partial\phi}{\partial z} = 0, \, z\in\partial\Omega. |
| </math></center> | | </math></center> |
| The equation is subject to some radiation conditions at infinity. We usually assume that
| | |
| there is an incident wave <math>\phi^{\mathrm{{In}}}\,</math>
| | {{incident plane wave}} |
| is a plane wave travelling in the <math>x</math> direction
| | |
| <center><math>
| | {{sommerfeld radiation condition two dimensions}} |
| \phi^{\mathrm{{In}}}({r},z)=Ae^{k_0 x}\frac{\cos k_0(z+h)}{\cos k_0 h}
| |
| </math></center>
| |
| where <math>A</math> is the wave amplitude and <math>k_0</math> is
| |
| the positive imaginary solution of the [[Dispersion Relation for a Free Surface]].
| |
| We also have to apply the [[Sommerfeld Radiation Condition]] as <math>\left|\mathbf{r}\right|\rightarrow
| |
| \infty</math>.
| |
| In two-dimensions the condition is
| |
| <center><math>
| |
| \left( \frac{\partial}{\partial|x|} -k_0\right)
| |
| (\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.}
| |
| </math></center>
| |
Revision as of 06:27, 24 August 2008
The Standard Linear Wave Scattering Problem
in Finite Depth for a fixed body in two dimensions is
[math]\displaystyle{
\nabla^{2}\phi=0, \, -h\lt z\lt 0,\,\,\,\mathbf{x}\notin \Omega
}[/math]
[math]\displaystyle{
\frac{\partial\phi}{\partial z}=0, \, z=-h,
}[/math]
[math]\displaystyle{
\frac{\partial\phi}{\partial z} = \alpha \phi,\,z=0,\,\,\mathbf{x}\notin\Omega,
}[/math]
[math]\displaystyle{
\frac{\partial\phi}{\partial z} = 0, \, z\in\partial\Omega.
}[/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.
