Sommerfeld Radiation Condition

This is a condition for the Frequency Domain Problem that the scattered wave is only outgoing at infinity.

In two-dimensions the condition is

[math]\displaystyle{ \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{ \sqrt{|\mathbf{r}|}\left( \frac{\partial}{\partial|\mathbf{r}|}-{i}k\right) (\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|\mathbf{r}|\rightarrow\infty\mathrm{.} }[/math]