Sommerfeld Radiation Condition
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{ \sqrt{|{x}|}\left( \frac{\partial}{\partial|x|}+{i}k\right) (\phi-\phi^{\mathrm{{In}}})=0,\;\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{ |\mathbf{r}|\left( \frac{\partial}{\partial|\mathbf{r}|}+{i}k\right) (\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|\mathbf{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{ \sqrt{|{x}|} \left( \frac{\partial}{\partial|x|}-{i}k\right) (\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.} }[/math]
and in three-dimensions
[math]\displaystyle{ |\mathbf{r}|\left( \frac{\partial}{\partial|\mathbf{r}|}-{i}k\right) (\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|\mathbf{r}|\rightarrow\infty\mathrm{.} }[/math]
