Sommerfeld Radiation Condition

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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 $\exp (i\omega t)\,$ or $\exp (-i\omega t)\,$. Assuming the former (which is the standard convention on this wiki). In two dimensions the condition is

$\left( \frac{\partial}{\partial|x|}+\mathrm{i}k\right) (\phi-\phi^{\mathrm{{In}}})\rightarrow0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.}$

where $\phi^{\mathrm{{In}}}$ is the incident potential and $k$ is the wave number.

In three dimensions the condition is

$r^{1/2}\left( \frac{\partial}{\partial r}+\mathrm{i}k\right) (\phi-\phi^{\mathrm{{In}}})\rightarrow0,\;\mathrm{{as\;}}r\rightarrow\infty\mathrm{.}$

If the time dependence is assumed to be $\exp (-i\omega t)\,$, then we have in two dimensions

$\left( \frac{\partial}{\partial|x|}-\mathrm{i}k\right) (\phi-\phi^{\mathrm{{In}}})\rightarrow0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.}$

and in three dimensions

$r^{1/2}\left( \frac{\partial}{\partial r}-\mathrm{i}k\right) (\phi-\phi^{\mathrm{{In}}})\rightarrow0,\;\mathrm{{as\;}}r\rightarrow\infty\mathrm{.}$