S wave

For the lowest-energy electronic wavefunction in atomic physics, seeatomic orbital.For the S wave on an electrocardiogram, seeQRS complex.

Plane shear wave

Propagation of a spherical S wave in a 2d grid (empirical model)

Inseismologyand other areas involving elastic waves,S waves,secondary waves,orshear waves(sometimes calledelastic S waves) are a type ofelastic waveand are one of the two main types of elasticbody waves,so named because they move through the body of an object, unlikesurface waves.

S waves aretransverse waves,meaning that the direction ofparticlemovement of an S wave is perpendicular to the direction of wave propagation, and the main restoring force comes fromshear stress.Therefore, S waves cannot propagate in liquidswith zero (or very low)viscosity;however, they may propagate in liquids with high viscosity.

The namesecondary wavecomes from the fact that they are the second type of wave to be detected by an earthquakeseismograph,after thecompressionalprimary wave, orP wave,because S waves travel more slowly in solids. Unlike P waves, S waves cannot travel through the moltenouter coreof the Earth, and this causes ashadow zonefor S waves opposite to their origin. They can still propagate through the solidinner core:when a P wave strikes the boundary of molten and solid cores at an oblique angle, S waves will form and propagate in the solid medium. When these S waves hit the boundary again at an oblique angle, they will in turn create P waves that propagate through the liquid medium. This property allowsseismologiststo determine some physical properties of the Earth's inner core.

History

In 1830, the mathematicianSiméon Denis Poissonpresented to theFrench Academy of Sciencesan essay ( "memoir" ) with a theory of the propagation of elastic waves in solids. In his memoir, he states that an earthquake would produce two different waves: one having a certain speed$a$and the other having a speed${\frac {a}{\sqrt {3}}}$.At a sufficient distance from the source, when they can be consideredplane wavesin the region of interest, the first kind consists of expansions and compressions in the direction perpendicular to the wavefront (that is, parallel to the wave's direction of motion); while the second consists of stretching motions occurring in directions parallel to the front (perpendicular to the direction of motion).

Theory

Isotropic medium

Velocity of seismic waves in the Earth versus depth. The negligible S-wave velocity in the outer core occurs because it is liquid, while in the solid inner core the S-wave velocity is non-zero.

For the purpose of this explanation, a solid medium is consideredisotropicif itsstrain (deformation)in response tostressis the same in all directions. Let${\boldsymbol {u}}=(u_{1},u_{2},u_{3})$be the displacementvectorof a particle of such a medium from its "resting" position${\boldsymbol {x}}=(x_{1},x_{2},x_{3})$due elastic vibrations, understood to be afunctionof the rest position${\boldsymbol {x}}$and time$t$.The deformation of the medium at that point can be described by thestrain tensor${\boldsymbol {e}}$,the 3×3 matrix whose elements are $$e_{ij}={\tfrac {1}{2}}\left(\partial _{i}u_{j}+\partial _{j}u_{i}\right)$$

where$\partial _{i}$denotes partial derivative with respect to position coordinate$x_{i}$.The strain tensor is related to the 3×3stress tensor${\boldsymbol {\tau }}$by the equation $$\tau _{ij}=\lambda \delta _{ij}\sum _{k}e_{kk}+2\mu e_{ij}$$

Here$\delta _{ij}$is theKronecker delta(1 if$i=j$,0 otherwise) and$\lambda$and$\mu$are theLamé parameters($\mu$being the material'sshear modulus). It follows that $$\tau _{ij}=\lambda \delta _{ij}\sum _{k}\partial _{k}u_{k}+\mu \left(\partial _{i}u_{j}+\partial _{j}u_{i}\right)$$

FromNewton's law of inertia,one also gets $$\rho \partial _{t}^{2}u_{i}=\sum _{j}\partial _{j}\tau _{ij}$$ where$\rho$is thedensity(mass per unit volume) of the medium at that point, and$\partial _{t}$denotes partial derivative with respect to time. Combining the last two equations one gets theseismic wave equation in homogeneous media $$\rho \partial _{t}^{2}u_{i}=\lambda \partial _{i}\sum _{k}\partial _{k}u_{k}+\mu \sum _{j}{\bigl (}\partial _{i}\partial _{j}u_{j}+\partial _{j}\partial _{j}u_{i}{\bigr )}$$

Using thenabla operatornotation ofvector calculus,$\nabla =(\partial _{1},\partial _{2},\partial _{3})$,with some approximations, this equation can be written as $$\rho \partial _{t}^{2}{\boldsymbol {u}}=\left(\lambda +2\mu \right)\nabla \left(\nabla \cdot {\boldsymbol {u}}\right)-\mu \nabla \times \left(\nabla \times {\boldsymbol {u}}\right)$$

Taking thecurlof this equation and applying vector identities, one gets $$\partial _{t}^{2}(\nabla \times {\boldsymbol {u}})={\frac {\mu }{\rho }}\nabla ^{2}\left(\nabla \times {\boldsymbol {u}}\right)$$

This formula is thewave equationapplied to the vector quantity$\nabla \times {\boldsymbol {u}}$,which is the material's shear strain. Its solutions, the S waves, arelinear combinationsofsinusoidalplane wavesof variouswavelengthsand directions of propagation, but all with the same speed${\textstyle \beta ={\sqrt {\mu /\rho }}}$.Assuming that the medium of propagation is linear, elastic, isotropic, and homogeneous, this equation can be rewritten as$\mu =\rho \beta ^{2}=\rho \omega ^{2}/k^{2}$whereωis the angular frequency andkis the wavenumber. Thus,$\beta =\omega /k$.

Taking thedivergenceof seismic wave equation in homogeneous media, instead of the curl, yields a wave equation describing propagation of the quantity$\nabla \cdot {\boldsymbol {u}}$,which is the material's compression strain. The solutions of this equation, the P waves, travel at the speed${\textstyle \alpha ={\sqrt {(\lambda +2\mu )/\rho }}}$that is more than twice the speed$\beta$of S waves.

Thesteady stateSH waves are defined by theHelmholtz equation $$\left(\nabla ^{2}+k^{2}\right){\boldsymbol {u}}=0$$ wherekis the wave number.

S waves in viscoelastic materials

Similar to in an elastic medium, in aviscoelasticmaterial, the speed of a shear wave is described by a similar relationship$c(\omega )=\omega /k(\omega )={\sqrt {\mu (\omega )/\rho }}$,however, here,$\mu$is a complex, frequency-dependent shear modulus and$c(\omega )$is the frequency dependent phase velocity.One common approach to describing the shear modulus in viscoelastic materials is through theVoigt Modelwhich states:$\mu (\omega )=\mu _{0}+i\omega \eta$,where$\mu _{0}$is the stiffness of the material and$\eta$is the viscosity.

S wave technology

Magnetic resonance elastography

Magnetic resonance elastography(MRE) is a method for studying the properties of biological materials in living organisms by propagating shear waves at desired frequencies throughout the desired organic tissue.This method uses a vibrator to send the shear waves into the tissue andmagnetic resonance imagingto view the response in the tissue.The measured wave speed and wavelengths are then measured to determine elastic properties such as theshear modulus.MRE has seen use in studies of a variety of human tissues including liver, brain, and bone tissues.

See also

• Earthquake Early Warning (Japan)
• Lamb waves
• Longitudinal wave
• Love wave
• P wave
• Rayleigh wave
• Seismic wave
• Shear wave splitting

Further reading

• Shearer, Peter (1999).Introduction to Seismology(1st ed.). Cambridge University Press.ISBN0-521-66023-8.
• Aki, Keiiti;Richards, Paul G.(2002).Quantitative Seismology(2nd ed.). University Science Books.ISBN0-935702-96-2.
• Fowler, C. M. R.(1990).The solid earth.Cambridge, UK: Cambridge University Press.ISBN0-521-38590-3.S-wave.