Sommerfeld Radiation Condition: Difference between revisions
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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 | 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}}}) | (\phi-\phi^{\mathrm{{In}}})\rightarrow0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.} | ||
</math> | </math> | ||
</center> | </center> | ||
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is the wave number. | is the wave number. | ||
In three | In three dimensions the condition is | ||
<center> | <center> | ||
<math> | <math> | ||
r^{1/2}\left( \frac{\partial}{\partial r}+\mathrm{i}k\right) | |||
(\phi-\phi^{\mathrm{{In}}}) | (\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 | 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}}}) | (\phi-\phi^{\mathrm{{In}}})\rightarrow0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.} | ||
</math> | </math> | ||
</center> | </center> | ||
and in three | and in three dimensions | ||
<center> | <center> | ||
<math> | <math> | ||
r^{1/2}\left( \frac{\partial}{\partial r}-\mathrm{i}k\right) | |||
(\phi-\phi^{\mathrm{{In}}}) | (\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]
