Difference between revisions of "Sommerfeld Radiation Condition"

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{{complete pages}}
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This is a condition for the [[Frequency Domain Problem]] that the scattered wave is only
 
This is a condition for the [[Frequency Domain Problem]] that the scattered wave is only
 
outgoing at infinity. It depends on the convention regarding whether the time dependence
 
outgoing at infinity. It depends on the convention regarding whether the time dependence
is <math>\exp (i\omega t)\,</math> or <math>\exp (-i\omega t)\,</math>
+
is <math>\exp (i\omega t)\,</math> or <math>\exp (-i\omega t)\,</math>.
Assuming the former (which is the standard convention on this wiki)  
+
Assuming the former (which is the standard convention on this wiki).
In two-dimensions the condition is  
+
In two dimensions the condition is  
 
<center>
 
<center>
 
<math>
 
<math>
\left(  \frac{\partial}{\partial|x|}+{i}k\right)
+
\left(  \frac{\partial}{\partial|x|}+\mathrm{i}k\right)
(\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.}
+
(\phi-\phi^{\mathrm{{In}}})\rightarrow0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.}
 
</math>
 
</math>
 
</center>
 
</center>
Line 13: Line 15:
 
is the wave number.
 
is the wave number.
  
In three-dimensions the condition is  
+
In three dimensions the condition is  
 
<center>
 
<center>
 
<math>
 
<math>
\sqrt{|\mathbf{r}|}\left(  \frac{\partial}{\partial|\mathbf{r}|}+{i}k\right)
+
r^{1/2}\left(  \frac{\partial}{\partial r}+\mathrm{i}k\right)
(\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|\mathbf{r}|\rightarrow\infty\mathrm{.}
+
(\phi-\phi^{\mathrm{{In}}})\rightarrow0,\;\mathrm{{as\;}}r\rightarrow\infty\mathrm{.}
 
</math>
 
</math>
 
</center>
 
</center>
  
If the time dependence is assumed to be <math>\exp (-i\omega t)\,</math> then we  
+
If the time dependence is assumed to be <math>\exp (-i\omega t)\,</math>, then we  
have in two-dimensions  
+
have in two dimensions  
 
<center>
 
<center>
 
<math>
 
<math>
\left(  \frac{\partial}{\partial|x|}-{i}k\right)
+
\left(  \frac{\partial}{\partial|x|}-\mathrm{i}k\right)
(\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.}
+
(\phi-\phi^{\mathrm{{In}}})\rightarrow0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.}
 
</math>
 
</math>
 
</center>
 
</center>
and in three-dimensions  
+
and in three dimensions  
 
<center>
 
<center>
 
<math>
 
<math>
\sqrt{|\mathbf{r}|}\left(  \frac{\partial}{\partial|\mathbf{r}|}-{i}k\right)
+
r^{1/2}\left(  \frac{\partial}{\partial r}-\mathrm{i}k\right)
(\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|\mathbf{r}|\rightarrow\infty\mathrm{.}
+
(\phi-\phi^{\mathrm{{In}}})\rightarrow0,\;\mathrm{{as\;}}r\rightarrow\infty\mathrm{.}
 
</math>
 
</math>
 
</center>
 
</center>
  
 
[[Category:Linear Water-Wave Theory]]
 
[[Category:Linear Water-Wave Theory]]

Latest revision as of 04:55, 4 September 2012


This is a condition for the Frequency Domain Problem that the scattered wave is only outgoing at infinity. It depends on the convention regarding whether the time dependence is [math]\displaystyle{ \exp (i\omega t)\, }[/math] or [math]\displaystyle{ \exp (-i\omega t)\, }[/math]. Assuming the former (which is the standard convention on this wiki). In two dimensions the condition is

[math]\displaystyle{ \left( \frac{\partial}{\partial|x|}+\mathrm{i}k\right) (\phi-\phi^{\mathrm{{In}}})\rightarrow0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.} }[/math]

where [math]\displaystyle{ \phi^{\mathrm{{In}}} }[/math] is the incident potential and [math]\displaystyle{ k }[/math] is the wave number.

In three dimensions the condition is

[math]\displaystyle{ r^{1/2}\left( \frac{\partial}{\partial r}+\mathrm{i}k\right) (\phi-\phi^{\mathrm{{In}}})\rightarrow0,\;\mathrm{{as\;}}r\rightarrow\infty\mathrm{.} }[/math]

If the time dependence is assumed to be [math]\displaystyle{ \exp (-i\omega t)\, }[/math], then we have in two dimensions

[math]\displaystyle{ \left( \frac{\partial}{\partial|x|}-\mathrm{i}k\right) (\phi-\phi^{\mathrm{{In}}})\rightarrow0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.} }[/math]

and in three dimensions

[math]\displaystyle{ r^{1/2}\left( \frac{\partial}{\partial r}-\mathrm{i}k\right) (\phi-\phi^{\mathrm{{In}}})\rightarrow0,\;\mathrm{{as\;}}r\rightarrow\infty\mathrm{.} }[/math]