Difference between revisions of "Template:Radiation condition for diffracted potential"

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<center><math>
 
<center><math>
 
\frac{\partial}{\partial x} \left(\phi^{\mathrm{D}}-\phi^{\rm
 
\frac{\partial}{\partial x} \left(\phi^{\mathrm{D}}-\phi^{\rm
I} \right) \pm ik\left( \phi^{\mathrm{D}}-\phi^{\rm I}\right)
+
I} \right) \pm k_0\left( \phi^{\mathrm{D}}-\phi^{\rm I}\right)
 
= 0
 
= 0
 
,\,\,\mathrm{as}
 
,\,\,\mathrm{as}
\,\,x\rightarrow\pm\infty,
+
\,\,x\rightarrow\infty.
 
</math></center>
 
</math></center>
where
+
{{incident plane wave 2d definition}}
<math>k ,\,</math> is the wavenumber,
 
which is the positive real solution of the [[Dispersion Relation for a Free Surface]]
 
<center><math>
 
k\tanh(kh)=\omega^{2} \,
 
</math></center>
 
and <math>\phi^{\rm I}</math> is the incident wave given by
 
<center><math>
 
\phi^{\rm I}  = \frac{\cosh(k(z+h))}{\cosh(kh)} e^{-i kx} \,
 
</math></center>
 
(which has unit amplitude in potential) and
 
is travelling towards positive
 
infinity.
 

Latest revision as of 03:22, 26 November 2009

[math]\displaystyle{ \phi^{\mathrm{D}} }[/math] satisfies the Sommerfeld Radiation Condition

[math]\displaystyle{ \frac{\partial}{\partial x} \left(\phi^{\mathrm{D}}-\phi^{\rm I} \right) \pm k_0\left( \phi^{\mathrm{D}}-\phi^{\rm I}\right) = 0 ,\,\,\mathrm{as} \,\,x\rightarrow\infty. }[/math]

[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]