Dimensional regularization - Revision history

en>Omnipaedista: rm 'see also' section per WP:ALSO

2014-12-23T13:06:35Z

rm 'see also' section per WP:ALSO

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en>Xezbeth: dab

2014-02-08T12:02:41Z

dab

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193.173.39.55: /* References */

2014-02-03T16:55:57Z

References

← Older revision		Revision as of 18:55, 3 February 2014
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	~~57 yrs old Registered Nurse~~ (~~Mental Health~~ ) ~~Anibal Gartman~~ from ~~South Porcupine~~, ~~loves~~ to ~~spend time jet engines~~, ~~new launch~~ property ~~singapore~~ and ~~post cards~~. ~~Has completed~~ a ~~great around~~ the ~~world tour that included~~ ~~traveling~~ to the ~~Viñales Valley~~.<br><br>~~Also visit my web blog -~~ [~~http~~://~~help~~.~~ieg4~~.~~com~~/~~entries~~/~~77099207~~-10-~~Health~~-~~Tips~~-~~For~~-~~Individuals~~-~~Who~~-Do-~~Not~~-~~Want~~-To-~~Work~~-~~Out The Skywoods forum~~]		{{about\|\|the lowest energy electronic wavefunction in atomic physics\|atomic orbital\|the S wave on an electrocardiogram\|QRS complex}}
			{{Earthquakes}}
			{{Cleanup-reorganize\|date=January 2009}}
			{{More footnotes\|date=January 2009}}[[Image:Onde cisaillement impulsion 1d 30 petit.gif\|thumb\|305px\|Plane shear wave]]

			[[Image:Ondes cisaillement 2d 20 petit.gif\|thumb\|305px\|Propagation of a spherical S-wave in a 2d grid (empirical model)]]

			A type of [[Linear elasticity#Elastic wave\|elastic wave]], the '''S-wave''', '''secondary wave''', or '''shear wave''' (sometimes called an '''elastic S-wave''') is one of the two main types of elastic [[body wave (seismology)\|body wave]]s, so named because they move through the body of an object, unlike [[surface wave]]s.

			The S-wave moves as a shear or [[transverse wave]], so motion is perpendicular to the direction of wave propagation. The wave moves through elastic media, and the main restoring force comes from shear effects. These waves do not diverge, and they obey the continuity equation for incompressible media:

			:<math>\nabla \cdot \mathbf{u} = 0</math>

			[[Image:Earthquake wave shadow zone.svg\|thumb\|150px\|The shadow zone of a [[P-wave]]. S-waves don't penetrate the outer core, so they're shadowed everywhere more than 104° away from the epicenter (from [[USGS]])]]

			Its name, S for secondary, comes from the fact that it is the second direct arrival on an earthquake [[seismogram]], after the [[compressional wave\|compressional]] primary wave, or [[P-wave]], because S-waves travel slower in rock. Unlike the P-wave, the S-wave cannot travel through the molten [[outer core]] of the Earth, and this causes a [[shadow zone]] for S-waves opposite to where they originate. They can still appear in the solid [[inner core]]: when a P-wave strikes the boundary of molten and solid cores, called the [[Lehmann discontinuity]], S-waves will then propagate in the solid medium. And when the S-waves hit the boundary again they will in turn create P-waves. This property allows [[seismology\|seismologists]] to determine the nature of the inner core.<ref name="UIC">{{cite web\|url=http://tigger.uic.edu/~rdemar/geol107/lect16.htm\|title=Lecture 16 Seismographs and the earth's interior\|last=University of Illinois at Chicago\|date=17 July 1997\|accessdate=8 June 2010}}</ref>

			As transverse waves, S-waves exhibit properties, such as [[Polarization (waves)\|polarization]] and [[birefringence]], much like other transverse waves. S-waves polarized in the horizontal plane are classified as SH-waves. If polarized in the vertical plane, they are classified as SV-waves. When an S- or P-wave strikes an interface at an angle other than 90 degrees, a phenomenon known as [[mode conversion]] occurs. As described above, if the interface is between a solid and liquid, S becomes P or vice versa. However, even if the interface is between two solid media, mode conversion results. If a P-wave strikes an interface, four propagation modes may result: reflected and transmitted P and reflected and transmitted SV. Similarly, if an SV-wave strikes an interface, the same four modes occur in different proportions. The exact amplitudes of all these waves are described by the [[Zoeppritz equations]], which in turn are solutions to the [[wave equation]].

			==Theory==
			The prediction of S-waves came out of theory in the 1800s. Starting with the [[Stress (mechanics)\|stress]]-[[Deformation (mechanics)\|strain]] relationship for an [[isotropic]] solid in [[Einstein notation]]:

			:<math>\tau_{ij}=\lambda\delta_{ij}e_{kk}+2\mu e_{ij}</math>

			where <math>\tau</math> is the stress, <math>\lambda</math> and <math>\mu</math> are the [[Lamé parameters]] (with <math>\mu</math> as the [[shear modulus]]), <math>\delta_{ij}</math> is the [[Kronecker delta]], and the strain tensor is defined

			:<math>e_{ij} = \frac{1}{2}\left( \partial_i u_j + \partial_j u_i \right)</math>

			for strain displacement u. Plugging the latter into the former yields:

			:<math>\tau_{ij} = \lambda\delta_{ij}\partial_ku_k + \mu \left( \partial_i u_j + \partial_j u_i \right)</math>

			[[Newton's 2nd law]] in this situation gives the ''homogeneous equation of motion'' for seismic wave propagation:

			:<math>\rho\frac{\partial^2u_i}{\partial t^2} = \partial_j\tau_{ij}</math>

			where <math>\rho</math> is the mass [[density]]. Plugging in the above stress tensor gives:

			:<math>\begin{align}
			\rho\frac{\partial^2 u_i}{\partial t^2}
			&= \partial_i\lambda\partial_k u_k + \partial_j\mu\left(\partial_i u_j + \partial_j u_i \right) \\
			&= \lambda\partial_i\partial_k u_k + \mu\partial_i\partial_j u_j + \mu\partial_j\partial_j u_i
			\end{align}</math>

			Applying vector identities and making certain approximations gives the seismic wave equation in homogeneous media:

			:<math>\rho \ddot{\boldsymbol{u}} = \left(\lambda + 2\mu \right)\nabla(\nabla\cdot\boldsymbol{u}) - \mu\nabla \times (\nabla \times \boldsymbol{u})</math>

			where [[Newton's notation]] has been used for the time derivative. Taking the [[Curl (mathematics)\|curl]] of this equation and applying vector identities eventually gives:

			:<math>\nabla^2(\nabla\times\boldsymbol{u}) - \frac{1}{\beta^2}\frac{\partial^2}{\partial t^2}\left(\nabla\times\boldsymbol{u}\right) = 0</math>

			which is simply the [[wave equation]] applied to the curl of u with a velocity <math>\beta</math> satisfying
			:<math>\beta^2 = \frac{\mu}{\rho}</math>

			This describes S-wave propagation. Taking the [[divergence]] of seismic wave equation in homogeneous media, instead of the curl, yields an equation describing P-wave propagation.
			The [[steady-state]] SH waves are defined by the [[Helmholtz equation]]
			:<math> (\nabla^2+k^2 )\boldsymbol{u}=0 </math> <ref>{{cite journal\|last=Sheikhhassani\|first=Ramtin\|journal=Scattering of a plane harmonic SH wave by multiple layered inclusions\|date=2013\|doi=10.1016/j.wavemoti.2013.12.002}}</ref>
			where k is the wave number.

			==See also==
			* [[Earthquake Early Warning (Japan)]]
			* [[Lamb waves]]
			* [[Longitudinal wave]]
			* [[Love wave]]
			* [[P-Wave]]
			* [[Rayleigh wave]]
			* [[Seismic wave]]
			* [[Shear wave splitting]]
			* [[Surface wave]]

			==References==
			{{Reflist}}

			==Further reading==
			*{{Cite book\|
			last=Shearer\|first=Peter\|
			year=1999\|
			title=Introduction to Seismology\|
			edition=1st\|
			publisher=Cambridge University Press\|
			isbn=0-521-66023-8}}
			*{{Cite book\|
			last1=Aki\|first1=Keiiti\|authorlink1=Keiiti Aki\|
			last2=Richards\|first2=Paul G.\|authorlink2=Paul G. Richards\|
			year=2002\|
			title=Quantitative Seismology\|
			edition=2nd\|
			publisher=University Science Books\|
			isbn=0-935702-96-2\|
			url=http://books.google.com/books?id=sRhawFG5_EcC&printsec=frontcover&dq=isbn:0935702962&hl=en&sa=X&ei=XjtQUryYLK-o4APyz4HAAg&ved=0CCAQ6AEwAA#v=onepage&q&f=false}}
			*{{Cite book\|
			last=Fowler\|first=C. M. R.\|
			year=1990\|
			title=The solid earth\|
			location=Cambridge, UK\|
			publisher=Cambridge University Press\|
			isbn=0-521-38590-3}}

			{{Geotechnical engineering}}

			[[Category:Waves]]
			[[Category:Seismology and earthquake terminology]]

en>Xqbot: r2.7.3) (Robot: Modifying ko:차원적 규칙화 to ko:차원 조절

2012-08-12T11:52:55Z

r2.7.3) (Robot: Modifying ko:차원적 규칙화 to ko:차원 조절

New page

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