Curve orientation - Revision history

en>Cydebot: Robot - Speedily moving category Orientation to :Category:Orientation (geometry) per CFDS.

2014-11-11T01:26:59Z

Robot - Speedily moving category Orientation to Category:Orientation (geometry) per CFDS.

← Older revision		Revision as of 03:26, 11 November 2014
Line 1:		Line 1:
	~~The '''Sommerfeld identity''' is a mathematical identity, due [[Arnold Sommerfeld]], used in the theory of propagation of waves,~~		Greetings! I am Myrtle Shroyer. To gather coins is what her family members and her appreciate. Hiring has been my profession for some time but I've currently applied for another one. My family members lives in Minnesota and my family members loves it.<br><br>Also visit my web blog: [http://www.animecontent.com/blog/124234 www.animecontent.com]

	~~:<math>~~
	~~\frac{{e^{ik R} }}~~
	~~{R} = \int\limits_0^\infty I_0(\lambda r) e^{ - \mu \left\| z \right\| } \frac{{\lambda d \lambda}}{{\mu}}~~
	~~</math>~~

	~~where~~
	~~:<math>~~
	~~\mu =~~
	~~\sqrt {\lambda ^2 - k^2 }~~
	~~</math>~~
	is ~~to be taken with positive real part, to ensure the convergence of the integral~~ and ~~its vanishing in the limit <math> z \rightarrow \pm \infty </math> and~~
	~~:<math>~~
	~~R^2=r^2+z^2~~
	~~</math>~~.
	Here, <math>R</math> is the distance from the origin while <math>r</math> is the distance from the central axis of a cylinder as in the <math>(r,\phi,z)</math> [[cylindrical coordinate system]]. The function <math>I_0</math> is a [[Bessel function]]. Here the notation for ~~Bessel functions follows the German convention, to be consistent with the original notation used by Sommerfeld~~. ~~In English literature~~ it ~~is more common to use~~

	~~:<math>I_n(\rho)=J_n(i \rho)</math>.~~

	~~This identity is known as the '''Sommerfeld Identity''' [Ref~~.~~1,Pg.242].~~

	~~An alternative form is~~

	:<~~math~~>
	~~\frac{{e^{ik_0 r} }}~~
	~~{r} = i\int\limits_0^\infty {dk_\rho \frac{{k_\rho }}~~
	~~{{k_z }}J_0 (k_\rho \rho )e^{ik_z \left\| z \right\|} }~~
	<~~/math~~>
	~~Where~~
	:~~<math>~~
	~~k_z=(k_0^2-k_\rho^2)^{1/2}~~
	~~</math>~~

	[~~Ref.2,Pg.66]. The notation used here is different form that above~~: ~~<math>r</math> is now the distance from the origin and <math>\rho<~~/~~math> is the axial distance in a cylindrical system defined as <math>(\rho,\phi,z)<~~/~~math>~~.

	~~The physical interpretation is that a spherical wave can be expanded into a summation~~
	~~of cylindrical waves in <math>\rho</math> direction, multiplied by a [[plane wave]] in the <math>z<~~/~~math> direction; see the [[Jacobi-Anger expansion]]. The summation has to be taken over all the wavenumbers <math>k_\rho<~~/~~math>~~.

	~~== References ==~~
	~~# Sommerfeld, A~~.~~,''Partial Differential Equations in Physics'',Academic Press,New York,1964~~
	~~# Chew, W.C.,''Waves and Fields in Inhomogenous Media'',Van Nostrand Reinhold,New York,1990~~
	~~<br>~~

	~~[[Category:Mathematical identities]~~]
	~~{{physics-stub}}~~

en>AnomieBOT: Dating maintenance tags: {{Unref}}

2013-09-05T23:55:42Z

Dating maintenance tags: {{Unref}}

← Older revision		Revision as of 01:55, 6 September 2013
Line 1:		Line 1:
	~~I would like~~ to ~~introduce myself~~ to ~~you~~, ~~I am Jayson Simcox but I don't like when people use my full title. Her family members lives~~ in ~~Ohio~~. ~~What me and my family adore~~ is ~~bungee leaping but I've been using on new things recently~~. ~~For many years she's been operating as a journey agent~~.<br><br>~~Here~~ is ~~my website~~ :: ~~clairvoyant psychic~~ ([~~http~~://~~chorokdeul~~.co.kr/~~index~~.~~php?document_srl~~=~~324263&mid~~=~~customer21 experienced~~])		The '''Sommerfeld identity''' is a mathematical identity, due [[Arnold Sommerfeld]], used in the theory of propagation of waves,

			:<math>
			\frac{{e^{ik R} }}
			{R} = \int\limits_0^\infty I_0(\lambda r) e^{ - \mu \left\| z \right\| } \frac{{\lambda d \lambda}}{{\mu}}
			</math>

			where
			:<math>
			\mu =
			\sqrt {\lambda ^2 - k^2 }
			</math>
			is to be taken with positive real part, to ensure the convergence of the integral and its vanishing in the limit <math> z \rightarrow \pm \infty </math> and
			:<math>
			R^2=r^2+z^2
			</math>.
			Here, <math>R</math> is the distance from the origin while <math>r</math> is the distance from the central axis of a cylinder as in the <math>(r,\phi,z)</math> [[cylindrical coordinate system]]. The function <math>I_0</math> is a [[Bessel function]]. Here the notation for Bessel functions follows the German convention, to be consistent with the original notation used by Sommerfeld. In English literature it is more common to use

			:<math>I_n(\rho)=J_n(i \rho)</math>.

			This identity is known as the '''Sommerfeld Identity''' [Ref.1,Pg.242].

			An alternative form is

			:<math>
			\frac{{e^{ik_0 r} }}
			{r} = i\int\limits_0^\infty {dk_\rho \frac{{k_\rho }}
			{{k_z }}J_0 (k_\rho \rho )e^{ik_z \left\| z \right\|} }
			</math>
			Where
			:<math>
			k_z=(k_0^2-k_\rho^2)^{1/2}
			</math>

			[Ref.2,Pg.66]. The notation used here is different form that above: <math>r</math> is now the distance from the origin and <math>\rho</math> is the axial distance in a cylindrical system defined as <math>(\rho,\phi,z)</math>.

			The physical interpretation is that a spherical wave can be expanded into a summation
			of cylindrical waves in <math>\rho</math> direction, multiplied by a [[plane wave]] in the <math>z</math> direction; see the [[Jacobi-Anger expansion]]. The summation has to be taken over all the wavenumbers <math>k_\rho</math>.

			== References ==
			# Sommerfeld, A.,''Partial Differential Equations in Physics'',Academic Press,New York,1964
			# Chew, W.C.,''Waves and Fields in Inhomogenous Media'',Van Nostrand Reinhold,New York,1990
			<br>

			[[Category:Mathematical identities]]
			{{physics-stub}}

en>Tommy2010: Reverted edits by 77.100.224.208 to last revision by 71.146.74.108 (HG)

2010-04-19T19:19:16Z

Reverted edits by 77.100.224.208 to last revision by 71.146.74.108 (HG)

New page

I would like to introduce myself to you, I am Jayson Simcox but I don't like when people use my full title. Her family members lives in Ohio. What me and my family adore is bungee leaping but I've been using on new things recently. For many years she's been operating as a journey agent.<br><br>Here is my website :: clairvoyant psychic ([http://chorokdeul.co.kr/index.php?document_srl=324263&mid=customer21 experienced])