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{{about||the lowest energy electronic wavefunction in atomic physics|atomic orbital|the S wave on an electrocardiogram|QRS complex}}
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{{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]]

Revision as of 14:02, 8 February 2014

In case you have a long run investment view and looking to enter the property market, a condominium may be a suitable alternative for you. Buyers and tenants of condos are typically those who take a liking to managed properties. So should you see a condo purchase as purely an funding, a rental will be extra interesting to those looking to hire as a result of it's managed by administration corporations.

Bartley Ridge is one MRT stop away from NEX @ Serangoon, which is Singapore 's largest shopping mall. All of your banking, eating and retail needs can be met at NEX mega mall! It is common data that in Singapore , properties which might be near prepare stations have the most important upswing throughout financial increase times, and have the least impact during financial lean instances. Prior to now few years, properties close to prepare stations have doubled in value! Properties that are further away additionally experience progress, but at a a lot slower price. William King is the director of Lahore Actual Estate & Lahore Property , Dubai Property Rentals & Dubai Rents , Property & Real Estate Listing & Classifieds and Concrete Suppliers & Concrete Enterprise Listing Selecting a Actual Property Agent

One of many causes for hiring a Warranwood property management firm is that they keep a digital network of tenants and property owners. This is The Skywoods forum beneficial for individuals in search of a great place to stay with reasonable lease and required facilities since by going by means of a management company they will discover property homeowners that meet their standards. Equally, property homeowners also get model tenants by hiring competent managers. Look out for initiatives in Tanjong Pagar & other rejuvenation areas beneath the Singapore masterplan. One such challenge coming up is Spottiswoode 18. In Singapore , there is a Vendor Stamp Obligation (SSD), levied on the vendor, if they sell their units inside the first 4 years of purchase. Sengkang condo opens for preview

Recording the bottom variety of new models being launched in August with only one,118 items on the market, the August URA figures moreover confirmed an expected dip with a complete of 1,539 models bought. According to PropNex, this was the lowest number of new personal residential transactions (including Executive Condominium ) ecotech @ sunview in 2012, at a drop of 25.5 p.c m-on-m, with sixty two p.c of the general models supplied within the Exterior Central Region. If initiatives akin to Eco and Riversails are able to hit the launch pad in September, new private house product sales is extra prone to rebound inside the non-public property market."

Designed to supply the best family-friendly ambiance that you could expect, Riverbank @ Fernvale supplies all the fashionable apartment services and leisure features that residents are on the lookout for. Whether you want to relocate and settle with your family into a major space of the island that is rapidly gaining prominence, or whether you might be an investor with a watch for an incredible funding property, Riverbank @ Fernvale has precisely what you want.

Whether or not you're a bachelor who's residing in a rented home otherwise you own a small house, buying a new home all the time excites people. The house shopping for course of turns into emotional when Indians plan to purchase one. The following article offers info for real property patrons relating to sourcing skilled assistance in the Los Angeles luxurious residence market.Finding the perfect properties in Los Angeles generally is a actually difficult activity. You want local space data, market expertise, and professional steering. This strategically located condominium is surrounded by accessible amenities that can make you and your loved ones's life-style simpler and more enjoyable with the availability of buying facilities at each corner; you will never have a uninteresting day at Marine Blue. Farmer's Market Marketing campaign

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