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In seismology, S-waves, secondary waves, or shear waves (sometimes called an elastic S-wave) are a type of elastic wave, and are one of the two main types of elastic body waves, so named because they move through the body of an object, unlike surface waves.^[1]

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.^[2] These waves do not diverge, and they obey the continuity equation for incompressible media:

∇⋅u=0{displaystyle nabla cdot mathbf {u} =0}

Its name, S for secondary, comes from the fact that it is the second direct arrival on an earthquake seismogram, after the 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 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 allows seismologists to determine some physical properties of the inner core.^[3]

Theory

The prediction of S-waves came out of theory in the 1800s.^[4] Starting with the stress-strain relationship for an isotropic solid in Einstein notation:

τij=λδijekk+2μeij{displaystyle tau _{ij}=lambda delta _{ij}e_{kk}+2mu e_{ij}}

where τ{displaystyle tau } is the stress, λ{displaystyle lambda } and μ{displaystyle mu } are the Lamé parameters (with μ{displaystyle mu } as the shear modulus), δij{displaystyle delta _{ij}} is the Kronecker delta, and the strain tensor is defined

eij=12(∂iuj+∂jui){displaystyle e_{ij}={frac {1}{2}}left(partial _{i}u_{j}+partial _{j}u_{i}right)}

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

τij=λδij∂kuk+μ(∂iuj+∂jui){displaystyle tau _{ij}=lambda delta _{ij}partial _{k}u_{k}+mu left(partial _{i}u_{j}+partial _{j}u_{i}right)}

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

ρ∂2ui∂t2=∂jτij{displaystyle rho {frac {partial ^{2}u_{i}}{partial t^{2}}}=partial _{j}tau _{ij}}

where ρ{displaystyle rho } is the mass density. Plugging in the above stress tensor gives:

ρ∂2ui∂t2=∂iλ∂kuk+∂jμ(∂iuj+∂jui)=λ∂i∂kuk+μ∂i∂juj+μ∂j∂jui{displaystyle {begin{aligned}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{aligned}}}

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

ρu¨=(λ+2μ)∇(∇⋅u)−μ∇×(∇×u){displaystyle rho {ddot {boldsymbol {u}}}=left(lambda +2mu right)nabla (nabla cdot {boldsymbol {u}})-mu nabla times (nabla times {boldsymbol {u}})}

where Newton's notation has been used for the time derivative. Taking the curl of this equation and applying vector identities eventually gives:

∇2(∇×u)−1β2∂2∂t2(∇×u)=0{displaystyle nabla ^{2}(nabla times {boldsymbol {u}})-{frac {1}{beta ^{2}}}{frac {partial ^{2}}{partial t^{2}}}left(nabla times {boldsymbol {u}}right)=0}

which is simply the wave equation applied to the curl of u with a velocity β{displaystyle beta } satisfying

β2=μρ{displaystyle beta ^{2}={frac {mu }{rho }}}

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

(∇2+k2)u=0{displaystyle (nabla ^{2}+k^{2}){boldsymbol {u}}=0} ^[5]

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

Further reading

• 
  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.