Difference between revisions of "Sommerfeld Radiation Condition"
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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. It depends on the convention regarding | + | 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> | |
| + | Assuming the former (which is the standard convention on this wiki) | ||
In two-dimensions the condition is | In two-dimensions the condition is | ||
| − | + | <center> | |
<math> | <math> | ||
| − | \left( \frac{\partial}{\partial|x|} | + | \left( \frac{\partial}{\partial|x|}+{i}k\right) |
(\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.} | (\phi-\phi^{\mathrm{{In}}})=0,\;\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-dimensions the condition is | In three-dimensions the condition is | ||
| + | <center> | ||
| + | <math> | ||
| + | \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> | ||
| + | </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|}-{i}k\right) | ||
| + | (\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.} | ||
| + | </math> | ||
| + | </center> | ||
| + | and in three-dimensions | ||
| + | <center> | ||
<math> | <math> | ||
\sqrt{|\mathbf{r}|}\left( \frac{\partial}{\partial|\mathbf{r}|}-{i}k\right) | \sqrt{|\mathbf{r}|}\left( \frac{\partial}{\partial|\mathbf{r}|}-{i}k\right) | ||
(\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|\mathbf{r}|\rightarrow\infty\mathrm{.} | (\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|\mathbf{r}|\rightarrow\infty\mathrm{.} | ||
</math> | </math> | ||
| + | </center> | ||
[[Category:Linear Water-Wave Theory]] | [[Category:Linear Water-Wave Theory]] | ||
Revision as of 23:48, 10 September 2008
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|}+{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]
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|}-{i}k\right) (\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.} }[/math]
and in three-dimensions
[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]
