Sommerfeld Radiation Condition - Revision history

Fmontiel at 04:55, 4 September 2012

2012-09-04T04:55:38Z

← Older revision		Revision as of 04:55, 4 September 2012
Line 3:		Line 3:
	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 whether the time dependence		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>		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)		Assuming the former (which is the standard convention on this wiki).
	In two-dimensions the condition is		In two dimensions the condition is
	<center>		<center>
	<math>		<math>
	~~\sqrt{\|{x}\|}~~\left( \frac{\partial}{\partial\|x\|}+{i}k\right)		\left( \frac{\partial}{\partial\|x\|}+\mathrm{i}k\right)
	(\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}\|x\|\rightarrow\infty\mathrm{.}		(\phi-\phi^{\mathrm{{In}}})\rightarrow0,\;\mathrm{{as\;}}\|x\|\rightarrow\infty\mathrm{.}
	</math>		</math>
	</center>		</center>
Line 15:		Line 15:
	is the wave number.		is the wave number.

	In three-dimensions the condition is		In three dimensions the condition is
	<center>		<center>
	<math>		<math>
	~~\|\mathbf~~{r}\|\left( \frac{\partial}{\partial~~\|\mathbf{~~r}\|}+{i}k\right)		r^{1/2}\left( \frac{\partial}{\partial r}+\mathrm{i}k\right)
	(\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}~~\|\mathbf{~~r}\|\rightarrow\infty\mathrm{.}		(\phi-\phi^{\mathrm{{In}}})\rightarrow0,\;\mathrm{{as\;}}r\rightarrow\infty\mathrm{.}
	</math>		</math>
	</center>		</center>

	If the time dependence is assumed to be <math>\exp (-i\omega t)\,</math> then we		If the time dependence is assumed to be <math>\exp (-i\omega t)\,</math>, then we
	have in two-dimensions		have in two dimensions
	<center>		<center>
	<math>		<math>
	~~\sqrt{\|{x}\|}~~ \left( \frac{\partial}{\partial\|x\|}-{i}k\right)		\left( \frac{\partial}{\partial\|x\|}-\mathrm{i}k\right)
	(\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}\|x\|\rightarrow\infty\mathrm{.}		(\phi-\phi^{\mathrm{{In}}})\rightarrow0,\;\mathrm{{as\;}}\|x\|\rightarrow\infty\mathrm{.}
	</math>		</math>
	</center>		</center>
	and in three-dimensions		and in three dimensions
	<center>		<center>
	<math>		<math>
	~~\|\mathbf~~{r}\|\left( \frac{\partial}{\partial~~\|\mathbf{~~r}\|}-{i}k\right)		r^{1/2}\left( \frac{\partial}{\partial r}-\mathrm{i}k\right)
	(\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}~~\|\mathbf{~~r}\|\rightarrow\infty\mathrm{.}		(\phi-\phi^{\mathrm{{In}}})\rightarrow0,\;\mathrm{{as\;}}r\rightarrow\infty\mathrm{.}
	</math>		</math>
	</center>		</center>

	[[Category:Linear Water-Wave Theory]]		[[Category:Linear Water-Wave Theory]]

Mike smith: correction to Sommerfeld Rad Cond. admins please doublecheck.

2010-07-08T21:17:45Z

correction to Sommerfeld Rad Cond. admins please doublecheck.

← Older revision		Revision as of 21:17, 8 July 2010
Line 8:		Line 8:
	<center>		<center>
	<math>		<math>
	\left( \frac{\partial}{\partial\|x\|}+{i}k\right)		\sqrt{\|{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>
Line 18:		Line 18:
	<center>		<center>
	<math>		<math>
	~~\sqrt{~~\|\mathbf{r}\|}\left( \frac{\partial}{\partial\|\mathbf{r}\|}+{i}k\right)		\|\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>
Line 27:		Line 27:
	<center>		<center>
	<math>		<math>
	\left( \frac{\partial}{\partial\|x\|}-{i}k\right)		\sqrt{\|{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>
Line 34:		Line 34:
	<center>		<center>
	<math>		<math>
	~~\sqrt{~~\|\mathbf{r}\|}\left( \frac{\partial}{\partial\|\mathbf{r}\|}-{i}k\right)		\|\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>

Meylan at 19:15, 8 February 2010

2010-02-08T19:15:16Z

← Older revision		Revision as of 19:15, 8 February 2010
Line 1:		Line 1:
			{{complete pages}}

	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 whether the time dependence		outgoing at infinity. It depends on the convention regarding whether the time dependence

Meylan at 23:48, 10 September 2008

2008-09-10T23:48:23Z

← Older revision		Revision as of 23:48, 10 September 2008
Line 1:		Line 1:
	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\|}-{i}k\right)		\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]]

Meylan at 23:44, 10 September 2008

2008-09-10T23:44:08Z

← Older revision		Revision as of 23:44, 10 September 2008
Line 1:		Line 1:
	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.		outgoing at infinity. It depends on the convention regarding

	In two-dimensions the condition is		In two-dimensions the condition is

Meylan at 01:41, 2 June 2006

2006-06-02T01:41:05Z

← Older revision		Revision as of 01:41, 2 June 2006
Line 18:		Line 18:
	(\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>

			[[Category:Linear Water-Wave Theory]]

Meylan at 09:14, 22 May 2006

2006-05-22T09:14:09Z

New page

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

where <math>\phi^{\mathrm{{In}}}</math> is the incident potential and <math>k</math>
is the wave number.

In three-dimensions the condition is

<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>