Infinite Array Green Function - Revision history

Meylan: Reverted edits by DronvArroc (DronvArroc); changed back to last version by Meylan

2009-01-09T09:11:29Z

Reverted edits by DronvArroc (DronvArroc); changed back to last version by Meylan

← Older revision		Revision as of 09:11, 9 January 2009
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	[http://sakonze.qsh.eu/letoletot-1061.html sailor moon video game download] [http://bukgetert.0lx.net/article-1050.htm ludacris video stand up] [http://caraines.qsh.eu/pletre.htm eminem stan video download] [http://lofaako.strefa.pl/article384.html movies for the blind lyrics] [http://naceceli.qsh.eu/page749.html movie theater winghaven]
	=Introduction=		=Introduction=

DronvArroc: http://sakonze.qsh.eu/letoletot-1061.html

2009-01-09T06:08:11Z

http://sakonze.qsh.eu/letoletot-1061.html

← Older revision		Revision as of 06:08, 9 January 2009
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			[http://sakonze.qsh.eu/letoletot-1061.html sailor moon video game download] [http://bukgetert.0lx.net/article-1050.htm ludacris video stand up] [http://caraines.qsh.eu/pletre.htm eminem stan video download] [http://lofaako.strefa.pl/article384.html movies for the blind lyrics] [http://naceceli.qsh.eu/page749.html movie theater winghaven]
	=Introduction=		=Introduction=

Meylan: Reverted edits by DronvArroc (DronvArroc); changed back to last version by Meylan

2009-01-09T03:34:14Z

Reverted edits by DronvArroc (DronvArroc); changed back to last version by Meylan

← Older revision		Revision as of 03:34, 9 January 2009
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	[http://dronoubr.0lx.net/indeo-video-51.html indeo video 5.1] [http://tulilre.strefa.pl/2009-01-06-lovette-free-movies.html lovette free movies] [http://bukgetert.0lx.net/article-1.htm au pair movie in australia] [http://sakonze.qsh.eu/fizqasba-703.html michael jackson video release] [http://betheres.qsh.eu/comment-1719.htm soul coughing circle video]
	[http://licawol.strefa.pl/resource1056.htm pair hilton sex video] [http://flheddin.qsh.eu/sitemap.html http] [http://sakonze.qsh.eu/darpasro-1740.html mahnah mahnah video] [http://spfortne.qsh.eu/20081231-smart-movie-mobile.html smart movie mobile download] [http://bukgetert.0lx.net/article-901.htm summer cummings movies]
	=Introduction=		=Introduction=

DronvArroc: http://dronoubr.0lx.net/indeo-video-51.html

2009-01-08T22:49:31Z

http://dronoubr.0lx.net/indeo-video-51.html

← Older revision		Revision as of 22:49, 8 January 2009
Line 1:		Line 1:
			[http://dronoubr.0lx.net/indeo-video-51.html indeo video 5.1] [http://tulilre.strefa.pl/2009-01-06-lovette-free-movies.html lovette free movies] [http://bukgetert.0lx.net/article-1.htm au pair movie in australia] [http://sakonze.qsh.eu/fizqasba-703.html michael jackson video release] [http://betheres.qsh.eu/comment-1719.htm soul coughing circle video]
	[http://licawol.strefa.pl/resource1056.htm pair hilton sex video] [http://flheddin.qsh.eu/sitemap.html http] [http://sakonze.qsh.eu/darpasro-1740.html mahnah mahnah video] [http://spfortne.qsh.eu/20081231-smart-movie-mobile.html smart movie mobile download] [http://bukgetert.0lx.net/article-901.htm summer cummings movies]		[http://licawol.strefa.pl/resource1056.htm pair hilton sex video] [http://flheddin.qsh.eu/sitemap.html http] [http://sakonze.qsh.eu/darpasro-1740.html mahnah mahnah video] [http://spfortne.qsh.eu/20081231-smart-movie-mobile.html smart movie mobile download] [http://bukgetert.0lx.net/article-901.htm summer cummings movies]
	=Introduction=		=Introduction=

DronvArroc: http://licawol.strefa.pl/resource1056.htm

2009-01-08T16:10:20Z

http://licawol.strefa.pl/resource1056.htm

← Older revision		Revision as of 16:10, 8 January 2009
Line 1:		Line 1:
			[http://licawol.strefa.pl/resource1056.htm pair hilton sex video] [http://flheddin.qsh.eu/sitemap.html http] [http://sakonze.qsh.eu/darpasro-1740.html mahnah mahnah video] [http://spfortne.qsh.eu/20081231-smart-movie-mobile.html smart movie mobile download] [http://bukgetert.0lx.net/article-901.htm summer cummings movies]
	=Introduction=		=Introduction=

Meylan: Reverted edits by DronvArroc (DronvArroc); changed back to last version by Meylan

2008-12-22T00:03:43Z

Reverted edits by DronvArroc (DronvArroc); changed back to last version by Meylan

← Older revision		Revision as of 00:03, 22 December 2008
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	~~montrocc~~
	=Introduction=		=Introduction=

DronvArroc: letoc4tvip

2008-12-20T04:33:37Z

letoc4tvip

← Older revision		Revision as of 04:33, 20 December 2008
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			montrocc
	=Introduction=		=Introduction=

Meylan: Reverted edits by Ni0Ix2 (Ni0Ix2); changed back to last version by Meylan

2007-06-11T08:58:31Z

Reverted edits by Ni0Ix2 (Ni0Ix2); changed back to last version by Meylan

← Older revision		Revision as of 08:58, 11 June 2007
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	The bodies occupy <math>\Delta_m</math>, <math>-\infty < m < \infty</math>. Periodicity implies		The bodies occupy <math>\Delta_m</math>, <math>-\infty < m < \infty</math>. Periodicity implies
	that if <math>(x,y) \in \Delta_0</math>, then <math>(x,y ml) \in \Delta_m</math>,		that if <math>(x,y) \in \Delta_0</math>, then <math>(x,y+ml) \in \Delta_m</math>,
	<math>-\infty < m < \infty</math>.		<math>-\infty < m < \infty</math>.

Line 24:		Line 24:
	<center><math>		<center><math>
	\phi^{{\rm in}} = \frac{A}{k}		\phi^{{\rm in}} = \frac{A}{k}
	e^{ik (x\cos\theta y\sin\theta)}\,e^{kz},		e^{ik (x\cos\theta+y\sin\theta)}\,e^{kz},
	</math></center>		</math></center>
	where <math>A</math> is the dimensionless amplitude and <math>\theta</math> is the direction of		where <math>A</math> is the dimensionless amplitude and <math>\theta</math> is the direction of
Line 53:		Line 53:
	<center><math>		<center><math>
	\phi(\mathbf{x}) = \phi^{\rm in} (\mathbf{x},0)		\phi(\mathbf{x}) = \phi^{\rm in} (\mathbf{x},0)
	\sum_{m=-\infty}^{\infty} \int_{\Delta_{m}}		+\sum_{m=-\infty}^{\infty} \int_{\Delta_{m}}
	\left(G_{n_\xi}(\mathbf{x},\xi) \phi(\xi) - G(\mathbf{x},\xi) \phi_{n_\xi}(\xi) \right) d\xi		\left(G_{n_\xi}(\mathbf{x},\xi) \phi(\xi) - G(\mathbf{x},\xi) \phi_{n_\xi}(\xi) \right) d\xi
	</math></center>		</math></center>
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	<center><math>		<center><math>
	\phi(\mathbf{x}) = \phi^{\rm in}(\mathbf{x})		\phi(\mathbf{x}) = \phi^{\rm in}(\mathbf{x})
	\int_{\Delta_{0}}		+\int_{\Delta_{0}}
	\sum_{m=-\infty}^{\infty} \left(G^{\mathbf{P}}_{n_\xi}(\mathbf{x},\xi (0,ml,0))e^{im\sigma l} \phi(\xi)		\sum_{m=-\infty}^{\infty} \left(G^{\mathbf{P}}_{n_\xi}(\mathbf{x},\xi+(0,ml,0))e^{im\sigma l} \phi(\xi)
	- G^{\mathbf{P}} (\mathbf{x},\xi)e^{im\sigma l} \phi_{n_\xi}(\xi) \right) d\xi		- G^{\mathbf{P}} (\mathbf{x},\xi)e^{im\sigma l} \phi_{n_\xi}(\xi) \right) d\xi
	</math></center>		</math></center>
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	<center><math>		<center><math>
	G^{\mathbf{P}} (\mathbf{x};\xi)		G^{\mathbf{P}} (\mathbf{x};\xi)
	= \sum_{m=-\infty}^{\infty} G(\mathbf{x},\xi (0,ml,0))e^{im\sigma l}.		= \sum_{m=-\infty}^{\infty} G(\mathbf{x},\xi+(0,ml,0))e^{im\sigma l}.
	</math></center>		</math></center>

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	= \sum_{m=-\infty}^{\infty}		= \sum_{m=-\infty}^{\infty}
	\left[		\left[
	G\left(\mathbf{x};\xi) (0,ml)\right)		G\left(\mathbf{x};\xi)+(0,ml)\right)
	\frac{ik}{2} H_{0}		+ \frac{ik}{2} H_{0}
	\Big(k\sqrt{\left( X cl\right)^2 Y_{m}^2}\Big) e^{im\sigma l}		\Big(k\sqrt{\left( X+cl\right)^2 + Y_{m}^2}\Big) e^{im\sigma l}
	\right]		\right]
	-\sum_{m=-\infty}^{\infty}		-\sum_{m=-\infty}^{\infty}
	\frac{ik}{2}H_{0} \Big(k\sqrt{ (X cl)^2 Y_{m}^2 }\Big)		\frac{ik}{2}H_{0} \Big(k\sqrt{ (X+cl)^2 + Y_{m}^2 }\Big)
	e^{im\sigma l}		e^{im\sigma l}
	</math></center>		</math></center>
Line 114:		Line 114:
	<center><math>		<center><math>
	-\sum_{m=-\infty}^{\infty}		-\sum_{m=-\infty}^{\infty}
	\frac{ik}{2}H_{0} \Big(k\sqrt{ (X cl)^2 Y_{m}^2 }\Big)		\frac{ik}{2}H_{0} \Big(k\sqrt{ (X+cl)^2 + Y_{m}^2 }\Big)
	e^{im\sigma l}		e^{im\sigma l}
	-\frac{i}{l} \sum_{m=-\infty}^{\infty}		-\frac{i}{l} \sum_{m=-\infty}^{\infty}
	\frac{e^{ik \mu_{m} \|X c\| }\,e^{i \sigma_{m} Y_0}}{\mu_{m}}		\frac{e^{ik \mu_{m} \|X+c\| }\,e^{i \sigma_{m} Y_0}}{\mu_{m}}
	</math></center>		</math></center>
	[[Linton 1998]] where		[[Linton 1998]] where
	<math>\sigma_{m} = \sigma 2 m \pi/l</math> and		<math>\sigma_{m} = \sigma + 2 m \pi/l</math> and
	<center><math>		<center><math>
	\mu_m = \left[ 1-\left(\frac{\sigma_{m}}{k}		\mu_m = \left[ 1-\left(\frac{\sigma_{m}}{k}
Line 132:		Line 132:
	G_{\mathbf{P}} (\mathbf{x};\xi)		G_{\mathbf{P}} (\mathbf{x};\xi)
	= \sum_{m=-\infty}^{\infty}		= \sum_{m=-\infty}^{\infty}
	\left[ G\left(\mathbf{x};\xi (0,ml)\right)		\left[ G\left(\mathbf{x};\xi+(0,ml)\right)
	\frac{ik}{2} H_{0} \Big(k \sqrt{\left( X cl\right)^2 Y_{m}^2}\Big)		+ \frac{ik}{2} H_{0} \Big(k \sqrt{\left( X+cl\right)^2 + Y_{m}^2}\Big)
	\right] e^{im\sigma l}		\right] e^{im\sigma l}
	-\frac{i}{l}\sum_{m=-\infty}^{\infty}		-\frac{i}{l}\sum_{m=-\infty}^{\infty}
	\frac{e^{ik \mu_{m}\|X cl\| }e^{i\sigma_{m}Y_0}}{\mu_{m}}.		\frac{e^{ik \mu_{m}\|X+cl\| }e^{i\sigma_{m}Y_0}}{\mu_{m}}.
	</math></center>		</math></center>
	The convergence of the two sums depends on the value		The convergence of the two sums depends on the value
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	\begin{matrix}		\begin{matrix}
	\sigma_{-M-1}<-k<\sigma_{-M},\\		\sigma_{-M-1}<-k<\sigma_{-M},\\
	\sigma_{N}<k<\sigma_{N 1}.		\sigma_{N}<k<\sigma_{N+1}.
	\end{matrix}		\end{matrix}
	\right\}		\right\}
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	These equations can be written as		These equations can be written as
	<center><math>		<center><math>
	\frac{l}{2\pi}\left(\sigma k-2\pi \right) < M < \frac{l}{2\pi}\left(		\frac{l}{2\pi}\left(\sigma+k-2\pi \right) < M < \frac{l}{2\pi}\left(
	\sigma k \right),		\sigma+k \right),
	</math></center>		</math></center>
	and		and
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	<center><math>		<center><math>
	\lim_{x\to\infty}\phi^{s} (\mathbf{x},0) =		\lim_{x\to\infty}\phi^{s} (\mathbf{x},0) =
	A_{m}^{ } e^{-ik\mu_{m}x} e^{i\sigma_{m}y},		A_{m}^{+} e^{-ik\mu_{m}x} e^{i\sigma_{m}y},
	</math></center>		</math></center>
	where <math>A_{m}^{ }</math> are		where <math>A_{m}^{+}</math> are
	<center><math>		<center><math>
	A_{m}^{ } = -\frac{i}{\mu_{m}l}\int_{\Delta_0}		A_{m}^{+} = -\frac{i}{\mu_{m}l}\int_{\Delta_0}
	e^{-ik\mu_{m}\xi }e^{-i\sigma_{m}\eta}		e^{-ik\mu_{m}\xi }e^{-i\sigma_{m}\eta}
	\left[ k\phi (\xi,0)		\left[ k\phi (\xi,0)
Line 266:		Line 266:
	the <math>m=0</math> mode is given by		the <math>m=0</math> mode is given by
	<center><math>		<center><math>
	T = 1 A_{0}^{ }		T = 1 + A_{0}^{+}
	= 1 - \frac{i}{\mu_{0}l} \int_{\Delta_0} e^{-ik(\xi\cos\theta		= 1 - \frac{i}{\mu_{0}l} \int_{\Delta_0} e^{-ik(\xi\cos\theta
	\eta\sin\theta)}\left[ k\phi(\xi,0)		+\eta\sin\theta)}\left[ k\phi(\xi,0)
	- w(\xi) \right]		- w(\xi) \right]
	d\xi.		d\xi.
Line 280:		Line 280:
	This gives us		This gives us
	<center><math>		<center><math>
	\cos\theta = \left( \|R\|^2 \|T\|^2 \right) \cos\theta		\cos\theta = \left( \|R\|^2+\|T\|^2 \right) \cos\theta
	\sum_{m=-M,\,m \neq 0}^{N}		+ \sum_{m=-M,\,m \neq 0}^{N}
	\left( \|A_{m}^{-}\|^2 \cos\psi_{m}^{-} \|A_{m}^{ }\|^2		\left( \|A_{m}^{-}\|^2 \cos\psi_{m}^{-} +\|A_{m}^{+}\|^2
	\cos\psi_{m}^{ } \right).		\cos\psi_{m}^{+} \right).
	</math></center>		</math></center>
	The energy balance equation can be used as an accuracy		The energy balance equation can be used as an accuracy

Ni0Ix2 at 01:25, 10 June 2007

2007-06-10T01:25:12Z

← Older revision		Revision as of 01:25, 10 June 2007
Line 16:		Line 16:

	The bodies occupy <math>\Delta_m</math>, <math>-\infty < m < \infty</math>. Periodicity implies		The bodies occupy <math>\Delta_m</math>, <math>-\infty < m < \infty</math>. Periodicity implies
	that if <math>(x,y) \in \Delta_0</math>, then <math>(x,y+ml) \in \Delta_m</math>,		that if <math>(x,y) \in \Delta_0</math>, then <math>(x,y ml) \in \Delta_m</math>,
	<math>-\infty < m < \infty</math>.		<math>-\infty < m < \infty</math>.

Line 24:		Line 24:
	<center><math>		<center><math>
	\phi^{{\rm in}} = \frac{A}{k}		\phi^{{\rm in}} = \frac{A}{k}
	e^{ik (x\cos\theta+y\sin\theta)}\,e^{kz},		e^{ik (x\cos\theta y\sin\theta)}\,e^{kz},
	</math></center>		</math></center>
	where <math>A</math> is the dimensionless amplitude and <math>\theta</math> is the direction of		where <math>A</math> is the dimensionless amplitude and <math>\theta</math> is the direction of
Line 53:		Line 53:
	<center><math>		<center><math>
	\phi(\mathbf{x}) = \phi^{\rm in} (\mathbf{x},0)		\phi(\mathbf{x}) = \phi^{\rm in} (\mathbf{x},0)
	+\sum_{m=-\infty}^{\infty} \int_{\Delta_{m}}		\sum_{m=-\infty}^{\infty} \int_{\Delta_{m}}
	\left(G_{n_\xi}(\mathbf{x},\xi) \phi(\xi) - G(\mathbf{x},\xi) \phi_{n_\xi}(\xi) \right) d\xi		\left(G_{n_\xi}(\mathbf{x},\xi) \phi(\xi) - G(\mathbf{x},\xi) \phi_{n_\xi}(\xi) \right) d\xi
	</math></center>		</math></center>
Line 61:		Line 61:
	<center><math>		<center><math>
	\phi(\mathbf{x}) = \phi^{\rm in}(\mathbf{x})		\phi(\mathbf{x}) = \phi^{\rm in}(\mathbf{x})
	+\int_{\Delta_{0}}		\int_{\Delta_{0}}
	\sum_{m=-\infty}^{\infty} \left(G^{\mathbf{P}}_{n_\xi}(\mathbf{x},\xi+(0,ml,0))e^{im\sigma l} \phi(\xi)		\sum_{m=-\infty}^{\infty} \left(G^{\mathbf{P}}_{n_\xi}(\mathbf{x},\xi (0,ml,0))e^{im\sigma l} \phi(\xi)
	- G^{\mathbf{P}} (\mathbf{x},\xi)e^{im\sigma l} \phi_{n_\xi}(\xi) \right) d\xi		- G^{\mathbf{P}} (\mathbf{x},\xi)e^{im\sigma l} \phi_{n_\xi}(\xi) \right) d\xi
	</math></center>		</math></center>
Line 69:		Line 69:
	<center><math>		<center><math>
	G^{\mathbf{P}} (\mathbf{x};\xi)		G^{\mathbf{P}} (\mathbf{x};\xi)
	= \sum_{m=-\infty}^{\infty} G(\mathbf{x},\xi+(0,ml,0))e^{im\sigma l}.		= \sum_{m=-\infty}^{\infty} G(\mathbf{x},\xi (0,ml,0))e^{im\sigma l}.
	</math></center>		</math></center>

Line 96:		Line 96:
	= \sum_{m=-\infty}^{\infty}		= \sum_{m=-\infty}^{\infty}
	\left[		\left[
	G\left(\mathbf{x};\xi)+(0,ml)\right)		G\left(\mathbf{x};\xi) (0,ml)\right)
	+ \frac{ik}{2} H_{0}		\frac{ik}{2} H_{0}
	\Big(k\sqrt{\left( X+cl\right)^2 + Y_{m}^2}\Big) e^{im\sigma l}		\Big(k\sqrt{\left( X cl\right)^2 Y_{m}^2}\Big) e^{im\sigma l}
	\right]		\right]
	-\sum_{m=-\infty}^{\infty}		-\sum_{m=-\infty}^{\infty}
	\frac{ik}{2}H_{0} \Big(k\sqrt{ (X+cl)^2 + Y_{m}^2 }\Big)		\frac{ik}{2}H_{0} \Big(k\sqrt{ (X cl)^2 Y_{m}^2 }\Big)
	e^{im\sigma l}		e^{im\sigma l}
	</math></center>		</math></center>
Line 114:		Line 114:
	<center><math>		<center><math>
	-\sum_{m=-\infty}^{\infty}		-\sum_{m=-\infty}^{\infty}
	\frac{ik}{2}H_{0} \Big(k\sqrt{ (X+cl)^2 + Y_{m}^2 }\Big)		\frac{ik}{2}H_{0} \Big(k\sqrt{ (X cl)^2 Y_{m}^2 }\Big)
	e^{im\sigma l}		e^{im\sigma l}
	-\frac{i}{l} \sum_{m=-\infty}^{\infty}		-\frac{i}{l} \sum_{m=-\infty}^{\infty}
	\frac{e^{ik \mu_{m} \|X+c\| }\,e^{i \sigma_{m} Y_0}}{\mu_{m}}		\frac{e^{ik \mu_{m} \|X c\| }\,e^{i \sigma_{m} Y_0}}{\mu_{m}}
	</math></center>		</math></center>
	[[Linton 1998]] where		[[Linton 1998]] where
	<math>\sigma_{m} = \sigma + 2 m \pi/l</math> and		<math>\sigma_{m} = \sigma 2 m \pi/l</math> and
	<center><math>		<center><math>
	\mu_m = \left[ 1-\left(\frac{\sigma_{m}}{k}		\mu_m = \left[ 1-\left(\frac{\sigma_{m}}{k}
Line 132:		Line 132:
	G_{\mathbf{P}} (\mathbf{x};\xi)		G_{\mathbf{P}} (\mathbf{x};\xi)
	= \sum_{m=-\infty}^{\infty}		= \sum_{m=-\infty}^{\infty}
	\left[ G\left(\mathbf{x};\xi+(0,ml)\right)		\left[ G\left(\mathbf{x};\xi (0,ml)\right)
	+ \frac{ik}{2} H_{0} \Big(k \sqrt{\left( X+cl\right)^2 + Y_{m}^2}\Big)		\frac{ik}{2} H_{0} \Big(k \sqrt{\left( X cl\right)^2 Y_{m}^2}\Big)
	\right] e^{im\sigma l}		\right] e^{im\sigma l}
	-\frac{i}{l}\sum_{m=-\infty}^{\infty}		-\frac{i}{l}\sum_{m=-\infty}^{\infty}
	\frac{e^{ik \mu_{m}\|X+cl\| }e^{i\sigma_{m}Y_0}}{\mu_{m}}.		\frac{e^{ik \mu_{m}\|X cl\| }e^{i\sigma_{m}Y_0}}{\mu_{m}}.
	</math></center>		</math></center>
	The convergence of the two sums depends on the value		The convergence of the two sums depends on the value
Line 169:		Line 169:
	\begin{matrix}		\begin{matrix}
	\sigma_{-M-1}<-k<\sigma_{-M},\\		\sigma_{-M-1}<-k<\sigma_{-M},\\
	\sigma_{N}<k<\sigma_{N+1}.		\sigma_{N}<k<\sigma_{N 1}.
	\end{matrix}		\end{matrix}
	\right\}		\right\}
Line 175:		Line 175:
	These equations can be written as		These equations can be written as
	<center><math>		<center><math>
	\frac{l}{2\pi}\left(\sigma+k-2\pi \right) < M < \frac{l}{2\pi}\left(		\frac{l}{2\pi}\left(\sigma k-2\pi \right) < M < \frac{l}{2\pi}\left(
	\sigma+k \right),		\sigma k \right),
	</math></center>		</math></center>
	and		and
Line 220:		Line 220:
	<center><math>		<center><math>
	\lim_{x\to\infty}\phi^{s} (\mathbf{x},0) =		\lim_{x\to\infty}\phi^{s} (\mathbf{x},0) =
	A_{m}^{+} e^{-ik\mu_{m}x} e^{i\sigma_{m}y},		A_{m}^{ } e^{-ik\mu_{m}x} e^{i\sigma_{m}y},
	</math></center>		</math></center>
	where <math>A_{m}^{+}</math> are		where <math>A_{m}^{ }</math> are
	<center><math>		<center><math>
	A_{m}^{+} = -\frac{i}{\mu_{m}l}\int_{\Delta_0}		A_{m}^{ } = -\frac{i}{\mu_{m}l}\int_{\Delta_0}
	e^{-ik\mu_{m}\xi }e^{-i\sigma_{m}\eta}		e^{-ik\mu_{m}\xi }e^{-i\sigma_{m}\eta}
	\left[ k\phi (\xi,0)		\left[ k\phi (\xi,0)
Line 266:		Line 266:
	the <math>m=0</math> mode is given by		the <math>m=0</math> mode is given by
	<center><math>		<center><math>
	T = 1 + A_{0}^{+}		T = 1 A_{0}^{ }
	= 1 - \frac{i}{\mu_{0}l} \int_{\Delta_0} e^{-ik(\xi\cos\theta		= 1 - \frac{i}{\mu_{0}l} \int_{\Delta_0} e^{-ik(\xi\cos\theta
	+\eta\sin\theta)}\left[ k\phi(\xi,0)		\eta\sin\theta)}\left[ k\phi(\xi,0)
	- w(\xi) \right]		- w(\xi) \right]
	d\xi.		d\xi.
Line 280:		Line 280:
	This gives us		This gives us
	<center><math>		<center><math>
	\cos\theta = \left( \|R\|^2+\|T\|^2 \right) \cos\theta		\cos\theta = \left( \|R\|^2 \|T\|^2 \right) \cos\theta
	+ \sum_{m=-M,\,m \neq 0}^{N}		\sum_{m=-M,\,m \neq 0}^{N}
	\left( \|A_{m}^{-}\|^2 \cos\psi_{m}^{-} +\|A_{m}^{+}\|^2		\left( \|A_{m}^{-}\|^2 \cos\psi_{m}^{-} \|A_{m}^{ }\|^2
	\cos\psi_{m}^{+} \right).		\cos\psi_{m}^{ } \right).
	</math></center>		</math></center>
	The energy balance equation can be used as an accuracy		The energy balance equation can be used as an accuracy

Meylan: Reverted edits by Sx9Mzf (Sx9Mzf); changed back to last version by Meylan

2007-04-11T13:24:14Z

Reverted edits by Sx9Mzf (Sx9Mzf); changed back to last version by Meylan

← Older revision		Revision as of 13:24, 11 April 2007
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	The bodies occupy <math>\Delta_m</math>, <math>-\infty < m < \infty</math>. Periodicity implies		The bodies occupy <math>\Delta_m</math>, <math>-\infty < m < \infty</math>. Periodicity implies
	that if <math>(x,y) \in \Delta_0</math>, then <math>(x,y ml) \in \Delta_m</math>,		that if <math>(x,y) \in \Delta_0</math>, then <math>(x,y+ml) \in \Delta_m</math>,
	<math>-\infty < m < \infty</math>.		<math>-\infty < m < \infty</math>.

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	<center><math>		<center><math>
	\phi^{{\rm in}} = \frac{A}{k}		\phi^{{\rm in}} = \frac{A}{k}
	e^{ik (x\cos\theta y\sin\theta)}\,e^{kz},		e^{ik (x\cos\theta+y\sin\theta)}\,e^{kz},
	</math></center>		</math></center>
	where <math>A</math> is the dimensionless amplitude and <math>\theta</math> is the direction of		where <math>A</math> is the dimensionless amplitude and <math>\theta</math> is the direction of
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	<center><math>		<center><math>
	\phi(\mathbf{x}) = \phi^{\rm in} (\mathbf{x},0)		\phi(\mathbf{x}) = \phi^{\rm in} (\mathbf{x},0)
	\sum_{m=-\infty}^{\infty} \int_{\Delta_{m}}		+\sum_{m=-\infty}^{\infty} \int_{\Delta_{m}}
	\left(G_{n_\xi}(\mathbf{x},\xi) \phi(\xi) - G(\mathbf{x},\xi) \phi_{n_\xi}(\xi) \right) d\xi		\left(G_{n_\xi}(\mathbf{x},\xi) \phi(\xi) - G(\mathbf{x},\xi) \phi_{n_\xi}(\xi) \right) d\xi
	</math></center>		</math></center>
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	<center><math>		<center><math>
	\phi(\mathbf{x}) = \phi^{\rm in}(\mathbf{x})		\phi(\mathbf{x}) = \phi^{\rm in}(\mathbf{x})
	\int_{\Delta_{0}}		+\int_{\Delta_{0}}
	\sum_{m=-\infty}^{\infty} \left(G^{\mathbf{P}}_{n_\xi}(\mathbf{x},\xi (0,ml,0))e^{im\sigma l} \phi(\xi)		\sum_{m=-\infty}^{\infty} \left(G^{\mathbf{P}}_{n_\xi}(\mathbf{x},\xi+(0,ml,0))e^{im\sigma l} \phi(\xi)
	- G^{\mathbf{P}} (\mathbf{x},\xi)e^{im\sigma l} \phi_{n_\xi}(\xi) \right) d\xi		- G^{\mathbf{P}} (\mathbf{x},\xi)e^{im\sigma l} \phi_{n_\xi}(\xi) \right) d\xi
	</math></center>		</math></center>
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	<center><math>		<center><math>
	G^{\mathbf{P}} (\mathbf{x};\xi)		G^{\mathbf{P}} (\mathbf{x};\xi)
	= \sum_{m=-\infty}^{\infty} G(\mathbf{x},\xi (0,ml,0))e^{im\sigma l}.		= \sum_{m=-\infty}^{\infty} G(\mathbf{x},\xi+(0,ml,0))e^{im\sigma l}.
	</math></center>		</math></center>

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	= \sum_{m=-\infty}^{\infty}		= \sum_{m=-\infty}^{\infty}
	\left[		\left[
	G\left(\mathbf{x};\xi) (0,ml)\right)		G\left(\mathbf{x};\xi)+(0,ml)\right)
	\frac{ik}{2} H_{0}		+ \frac{ik}{2} H_{0}
	\Big(k\sqrt{\left( X cl\right)^2 Y_{m}^2}\Big) e^{im\sigma l}		\Big(k\sqrt{\left( X+cl\right)^2 + Y_{m}^2}\Big) e^{im\sigma l}
	\right]		\right]
	-\sum_{m=-\infty}^{\infty}		-\sum_{m=-\infty}^{\infty}
	\frac{ik}{2}H_{0} \Big(k\sqrt{ (X cl)^2 Y_{m}^2 }\Big)		\frac{ik}{2}H_{0} \Big(k\sqrt{ (X+cl)^2 + Y_{m}^2 }\Big)
	e^{im\sigma l}		e^{im\sigma l}
	</math></center>		</math></center>
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	<center><math>		<center><math>
	-\sum_{m=-\infty}^{\infty}		-\sum_{m=-\infty}^{\infty}
	\frac{ik}{2}H_{0} \Big(k\sqrt{ (X cl)^2 Y_{m}^2 }\Big)		\frac{ik}{2}H_{0} \Big(k\sqrt{ (X+cl)^2 + Y_{m}^2 }\Big)
	e^{im\sigma l}		e^{im\sigma l}
	-\frac{i}{l} \sum_{m=-\infty}^{\infty}		-\frac{i}{l} \sum_{m=-\infty}^{\infty}
	\frac{e^{ik \mu_{m} \|X c\| }\,e^{i \sigma_{m} Y_0}}{\mu_{m}}		\frac{e^{ik \mu_{m} \|X+c\| }\,e^{i \sigma_{m} Y_0}}{\mu_{m}}
	</math></center>		</math></center>
	[[Linton 1998]] where		[[Linton 1998]] where
	<math>\sigma_{m} = \sigma 2 m \pi/l</math> and		<math>\sigma_{m} = \sigma + 2 m \pi/l</math> and
	<center><math>		<center><math>
	\mu_m = \left[ 1-\left(\frac{\sigma_{m}}{k}		\mu_m = \left[ 1-\left(\frac{\sigma_{m}}{k}
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	G_{\mathbf{P}} (\mathbf{x};\xi)		G_{\mathbf{P}} (\mathbf{x};\xi)
	= \sum_{m=-\infty}^{\infty}		= \sum_{m=-\infty}^{\infty}
	\left[ G\left(\mathbf{x};\xi (0,ml)\right)		\left[ G\left(\mathbf{x};\xi+(0,ml)\right)
	\frac{ik}{2} H_{0} \Big(k \sqrt{\left( X cl\right)^2 Y_{m}^2}\Big)		+ \frac{ik}{2} H_{0} \Big(k \sqrt{\left( X+cl\right)^2 + Y_{m}^2}\Big)
	\right] e^{im\sigma l}		\right] e^{im\sigma l}
	-\frac{i}{l}\sum_{m=-\infty}^{\infty}		-\frac{i}{l}\sum_{m=-\infty}^{\infty}
	\frac{e^{ik \mu_{m}\|X cl\| }e^{i\sigma_{m}Y_0}}{\mu_{m}}.		\frac{e^{ik \mu_{m}\|X+cl\| }e^{i\sigma_{m}Y_0}}{\mu_{m}}.
	</math></center>		</math></center>
	The convergence of the two sums depends on the value		The convergence of the two sums depends on the value
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	\begin{matrix}		\begin{matrix}
	\sigma_{-M-1}<-k<\sigma_{-M},\\		\sigma_{-M-1}<-k<\sigma_{-M},\\
	\sigma_{N}<k<\sigma_{N 1}.		\sigma_{N}<k<\sigma_{N+1}.
	\end{matrix}		\end{matrix}
	\right\}		\right\}
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	These equations can be written as		These equations can be written as
	<center><math>		<center><math>
	\frac{l}{2\pi}\left(\sigma k-2\pi \right) < M < \frac{l}{2\pi}\left(		\frac{l}{2\pi}\left(\sigma+k-2\pi \right) < M < \frac{l}{2\pi}\left(
	\sigma k \right),		\sigma+k \right),
	</math></center>		</math></center>
	and		and
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	<center><math>		<center><math>
	\lim_{x\to\infty}\phi^{s} (\mathbf{x},0) =		\lim_{x\to\infty}\phi^{s} (\mathbf{x},0) =
	A_{m}^{ } e^{-ik\mu_{m}x} e^{i\sigma_{m}y},		A_{m}^{+} e^{-ik\mu_{m}x} e^{i\sigma_{m}y},
	</math></center>		</math></center>
	where <math>A_{m}^{ }</math> are		where <math>A_{m}^{+}</math> are
	<center><math>		<center><math>
	A_{m}^{ } = -\frac{i}{\mu_{m}l}\int_{\Delta_0}		A_{m}^{+} = -\frac{i}{\mu_{m}l}\int_{\Delta_0}
	e^{-ik\mu_{m}\xi }e^{-i\sigma_{m}\eta}		e^{-ik\mu_{m}\xi }e^{-i\sigma_{m}\eta}
	\left[ k\phi (\xi,0)		\left[ k\phi (\xi,0)
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	the <math>m=0</math> mode is given by		the <math>m=0</math> mode is given by
	<center><math>		<center><math>
	T = 1 A_{0}^{ }		T = 1 + A_{0}^{+}
	= 1 - \frac{i}{\mu_{0}l} \int_{\Delta_0} e^{-ik(\xi\cos\theta		= 1 - \frac{i}{\mu_{0}l} \int_{\Delta_0} e^{-ik(\xi\cos\theta
	\eta\sin\theta)}\left[ k\phi(\xi,0)		+\eta\sin\theta)}\left[ k\phi(\xi,0)
	- w(\xi) \right]		- w(\xi) \right]
	d\xi.		d\xi.
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	This gives us		This gives us
	<center><math>		<center><math>
	\cos\theta = \left( \|R\|^2 \|T\|^2 \right) \cos\theta		\cos\theta = \left( \|R\|^2+\|T\|^2 \right) \cos\theta
	\sum_{m=-M,\,m \neq 0}^{N}		+ \sum_{m=-M,\,m \neq 0}^{N}
	\left( \|A_{m}^{-}\|^2 \cos\psi_{m}^{-} \|A_{m}^{ }\|^2		\left( \|A_{m}^{-}\|^2 \cos\psi_{m}^{-} +\|A_{m}^{+}\|^2
	\cos\psi_{m}^{ } \right).		\cos\psi_{m}^{+} \right).
	</math></center>		</math></center>
	The energy balance equation can be used as an accuracy		The energy balance equation can be used as an accuracy