Sommerfeld Radiation Condition: Difference between revisions
No edit summary |
No edit summary |
||
| (5 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
{{complete pages}} | |||
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]] | |||
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]
