Difference between revisions of "Sommerfeld Radiation Condition"
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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. | + | 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>. | |
| − | In two | + | Assuming the former (which is the standard convention on this wiki). |
| − | + | In two dimensions the condition is | |
| + | <center> | ||
<math> | <math> | ||
| − | \left( \frac{\partial}{\partial|x|} | + | \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> | |
where <math>\phi^{\mathrm{{In}}}</math> is the incident potential and <math>k</math> | where <math>\phi^{\mathrm{{In}}}</math> is the incident potential and <math>k</math> | ||
is the wave number. | is the wave number. | ||
| − | In three | + | In three dimensions the condition is |
| + | <center> | ||
| + | <math> | ||
| + | r^{1/2}\left( \frac{\partial}{\partial r}+\mathrm{i}k\right) | ||
| + | (\phi-\phi^{\mathrm{{In}}})\rightarrow0,\;\mathrm{{as\;}}r\rightarrow\infty\mathrm{.} | ||
| + | </math> | ||
| + | </center> | ||
| + | If the time dependence is assumed to be <math>\exp (-i\omega t)\,</math>, then we | ||
| + | have in two dimensions | ||
| + | <center> | ||
| + | <math> | ||
| + | \left( \frac{\partial}{\partial|x|}-\mathrm{i}k\right) | ||
| + | (\phi-\phi^{\mathrm{{In}}})\rightarrow0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.} | ||
| + | </math> | ||
| + | </center> | ||
| + | and in three dimensions | ||
| + | <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> | ||
[[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]
