Difference between revisions of "Template:Fixed body finite depth equations in two dimensions"
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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> | ||
+ | is a plane wave travelling in the <math>x</math> direction | ||
+ | <center><math> | ||
+ | \phi^{\mathrm{{In}}}({r},z)=Ae^{{\rm i}kx}\cosh k(z+h)\, | ||
+ | </math></center> | ||
+ | where <math>A</math> is the wave amplitude and <math>k</math> is the wavenumber which is | ||
+ | the positive real 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|}-{i}k\right) | ||
+ | (\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.} | ||
+ | </math></center> | ||
+ | where <math>\phi^{\mathrm{{In}}}</math> is the incident potential and <math>k</math> | ||
+ | is the wave number. |
Revision as of 02:59, 18 August 2008
The Standard Linear Wave Scattering Problem in Finite Depth for a fixed body in two dimensions is
The equation is subject to some radiation conditions at infinity. We usually assume that there is an incident wave [math]\displaystyle{ \phi^{\mathrm{{In}}}\, }[/math] is a plane wave travelling in the [math]\displaystyle{ x }[/math] direction
where [math]\displaystyle{ A }[/math] is the wave amplitude and [math]\displaystyle{ k }[/math] is the wavenumber which is the positive real solution of the Dispersion Relation for a Free Surface. We also have to apply the Sommerfeld Radiation Condition as [math]\displaystyle{ \left|\mathbf{r}\right|\rightarrow \infty }[/math].
In two-dimensions the condition is
where [math]\displaystyle{ \phi^{\mathrm{{In}}} }[/math] is the incident potential and [math]\displaystyle{ k }[/math] is the wave number.