Plane shear wave

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

Inseismology and other areas involving elastic waves,S waves,secondary waves, orshear waves (sometimes calledelastic S waves) are a type ofelastic wave and are one of the two main types of elasticbody waves, so named because they move through the body of an object, unlikesurface waves.^[1]

S waves aretransverse waves, meaning that the direction ofparticle movement of an S wave is perpendicular to the direction of wave propagation, and the main restoring force comes fromshear stress.^[2] Therefore, S waves cannot propagate in liquids^[3] with zero (or very low)viscosity; however, they may propagate in liquids with high viscosity.^[4][5] Similarly, S waves cannot travel through gases.

The namesecondary wave comes from the fact that they are the second type of wave to be detected by an earthquakeseismograph, after thecompressional primary wave, orP wave, because S waves travel more slowly in solids. Unlike P waves, S waves cannot travel through the moltenouter core of the Earth, and this causes ashadow zone for 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 allowsseismologists to determine some physical properties of the Earth's inner core.^[6]

In 1830, the mathematicianSiméon Denis Poisson presented to theFrench Academy of Sciences an 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${\displaystyle a}$ and the other having a speed${\displaystyle \frac{a}{\sqrt{3}}}$. At a sufficient distance from the source, when they can be consideredplane waves in 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).^[7]

For the purpose of this explanation, a solid medium is consideredisotropic if itsstrain (deformation) in response tostress is the same in all directions. Let${\displaystyle \mathit{u}=(u_1,u_2,u_3)}$ be the displacementvector of a particle of such a medium from its "resting" position${\displaystyle \mathit{x}=(x_1,x_2,x_3)}$ due elastic vibrations, understood to be afunction of the rest position${\displaystyle \mathit{x}}$ and time${\displaystyle t}$. The deformation of the medium at that point can be described by thestrain tensor${\displaystyle \mathit{e}}$, the 3×3 matrix whose elements are$${\displaystyle e_{ij}=\frac{1}{2}\left({\mathrm{\partial }}_i{u}_j+{\mathrm{\partial }}_j{u}_i\right)}$$

where${\displaystyle {\mathrm{\partial }}_i}$ denotes partial derivative with respect to position coordinate${\displaystyle x_i}$. The strain tensor is related to the 3×3stress tensor${\displaystyle \mathit{\tau }}$ by the equation$${\displaystyle {\tau }_{ij}=\lambda {\delta }_{ij}\sum _k{e}_{kk}+2\mu e_{ij}}$$

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

FromNewton's law of inertia, one also gets$${\displaystyle \rho {\mathrm{\partial }}_t^2{u}_i=\sum _j{\mathrm{\partial }}_j{\tau }_{ij}}$$where${\displaystyle \rho }$ is thedensity (mass per unit volume) of the medium at that point, and${\displaystyle {\mathrm{\partial }}_t}$ denotes partial derivative with respect to time. Combining the last two equations one gets theseismic wave equation in homogeneous media$${\displaystyle \rho {\mathrm{\partial }}_t^2{u}_i=\lambda {\mathrm{\partial }}_i\sum _k{\mathrm{\partial }}_k{u}_k+\mu \sum _j({\mathrm{\partial }}_i{\mathrm{\partial }}_j{u}_j+{\mathrm{\partial }}_j{\mathrm{\partial }}_j{u}_i)}$$

Using thenabla operator notation ofvector calculus,${\displaystyle \mathrm{\nabla }=({\mathrm{\partial }}_1,{\mathrm{\partial }}_2,{\mathrm{\partial }}_3)}$, with some approximations, this equation can be written as$${\displaystyle \rho {\mathrm{\partial }}_t^2\mathit{u}=\left(\lambda +2\mu \right)\mathrm{\nabla }\left(\mathrm{\nabla }\cdot \mathit{u}\right)-\mu \mathrm{\nabla }\times \left(\mathrm{\nabla }\times \mathit{u}\right)}$$

Taking thecurl of this equation and applying vector identities, one gets$${\displaystyle {\mathrm{\partial }}_t^2(\mathrm{\nabla }\times \mathit{u})=\frac{\mu }{\rho }{\mathrm{\nabla }}^2\left(\mathrm{\nabla }\times \mathit{u}\right)}$$

This formula is thewave equation applied to the vector quantity${\displaystyle \mathrm{\nabla }\times \mathit{u}}$, which is the material's shear strain. Its solutions, the S waves, arelinear combinations ofsinusoidalplane waves of variouswavelengths and directions of propagation, but all with the same speed$\beta =\sqrt{\mu /\rho }$. Assuming that the medium of propagation is linear, elastic, isotropic, and homogeneous, this equation can be rewritten as${\displaystyle \mu =\rho {\beta }^2=\rho {\omega }^2/k^2}$^[8] whereω is the angular frequency andk is the wavenumber. Thus,${\displaystyle \beta =\omega /k}$.

Taking thedivergence of seismic wave equation in homogeneous media, instead of the curl, yields a wave equation describing propagation of the quantity${\displaystyle \mathrm{\nabla }\cdot \mathit{u}}$, which is the material's compression strain. The solutions of this equation, the P waves, travel at the faster speed$\alpha =\sqrt{(\lambda +2\mu )/\rho }$.

Thesteady state SH waves are defined by theHelmholtz equation^[9]$${\displaystyle \left({\mathrm{\nabla }}^2+k^2\right)\mathit{u}=0}$$wherek is the wave number.

Similar to in an elastic medium, in aviscoelastic material, the speed of a shear wave is described by a similar relationship${\displaystyle c(\omega )=\omega /k(\omega )=\sqrt{\mu (\omega )/\rho }}$, however, here,${\displaystyle \mu }$ is a complex, frequency-dependent shear modulus and${\displaystyle c(\omega )}$ is the frequency dependent phase velocity.^[8] One common approach to describing the shear modulus in viscoelastic materials is through theVoigt Model which states:${\displaystyle \mu (\omega )={\mu }_0+i\omega \eta }$, where${\displaystyle {\mu }_0}$ is the stiffness of the material and${\displaystyle \eta }$ is the viscosity.^[8]

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.^[10] This method uses a vibrator to send the shear waves into the tissue andmagnetic resonance imaging to view the response in the tissue.^[11] 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.^[10]

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

• Shearer, Peter (1999).Introduction to Seismology (1st ed.). Cambridge University Press.ISBN 0-521-66023-8.
• Aki, Keiiti;Richards, Paul G. (2002).Quantitative Seismology (2nd ed.). University Science Books.ISBN 0-935702-96-2.
• Fowler, C. M. R. (1990).The solid earth. Cambridge, UK: Cambridge University Press.ISBN 0-521-38590-3.S-wave.

• Waves
• Seismology

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