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

$\sqrt{|{x}|}\left( \frac{\partial}{\partial|x|}+{i}k\right) (\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.}$

where $φ In$ is the incident potential and $k$ is the wave number.

In three-dimensions the condition is

$|\mathbf{r}|\left( \frac{\partial}{\partial|\mathbf{r}|}+{i}k\right) (\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|\mathbf{r}|\rightarrow\infty\mathrm{.}$

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

$\sqrt{|{x}|} \left( \frac{\partial}{\partial|x|}-{i}k\right) (\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.}$

and in three-dimensions

$|\mathbf{r}|\left( \frac{\partial}{\partial|\mathbf{r}|}-{i}k\right) (\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|\mathbf{r}|\rightarrow\infty\mathrm{.}$

Sommerfeld Radiation Condition

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