# S-wave

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

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:

${\displaystyle \nabla \cdot \mathbf {u} =0}$
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 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, 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 seismologists to determine the nature 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:

${\displaystyle \tau _{ij}=\lambda \delta _{ij}e_{kk}+2\mu e_{ij}}$

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

${\displaystyle e_{ij}={\frac {1}{2}}\left(\partial _{i}u_{j}+\partial _{j}u_{i}\right)}$

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

${\displaystyle \tau _{ij}=\lambda \delta _{ij}\partial _{k}u_{k}+\mu \left(\partial _{i}u_{j}+\partial _{j}u_{i}\right)}$

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

${\displaystyle \rho {\frac {\partial ^{2}u_{i}}{\partial t^{2}}}=\partial _{j}\tau _{ij}}$

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

{\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:

${\displaystyle \rho {\ddot {\boldsymbol {u}}}=\left(\lambda +2\mu \right)\nabla (\nabla \cdot {\boldsymbol {u}})-\mu \nabla \times (\nabla \times {\boldsymbol {u}})}$

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

${\displaystyle \nabla ^{2}(\nabla \times {\boldsymbol {u}})-{\frac {1}{\beta ^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\left(\nabla \times {\boldsymbol {u}}\right)=0}$

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

${\displaystyle \beta ^{2}={\frac {\mu }{\rho }}}$

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

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

where k is the wave number.