Plane shear wave
Propagation of a spherical Swave in a 2d grid (empirical model)
Swaves, secondary waves, or shear waves (sometimes called an elastic Swave) 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.
The Swave 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:

\nabla \cdot \mathbf{u} = 0
The shadow zone of a
Pwave. Swaves 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 Pwave, because Swaves travel slower in rock. Unlike the Pwave, the Swave cannot travel through the molten outer core of the Earth, and this causes a shadow zone for Swaves opposite to where they originate. They can still appear in the solid inner core: when a Pwave strikes the boundary of molten and solid cores, Swaves will then propagate in the solid medium. And when the Swaves hit the boundary again they will in turn create Pwaves. This property allows seismologists to determine the nature of the inner core.^{[1]}
As transverse waves, Swaves exhibit properties, such as polarization and birefringence, much like other transverse waves. Swaves polarized in the horizontal plane are classified as SHwaves. If polarized in the vertical plane, they are classified as SVwaves. When an S or Pwave 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 Pwave strikes an interface, four propagation modes may result: reflected and transmitted P and reflected and transmitted SV. Similarly, if an SVwave 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.
Contents

Theory 1

See also 2

References 3

Further reading 4
Theory
The prediction of Swaves came out of theory in the 1800s. Starting with the stressstrain relationship for an isotropic solid in Einstein notation:

\tau_{ij}=\lambda\delta_{ij}e_{kk}+2\mu e_{ij}
where \tau is the stress, \lambda and \mu are the Lamé parameters (with \mu as the shear modulus), \delta_{ij} is the Kronecker delta, and the strain tensor is defined

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:

\tau_{ij} = \lambda\delta_{ij}\partial_ku_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:

\rho\frac{\partial^2u_i}{\partial t^2} = \partial_j\tau_{ij}
where \rho is the mass density. Plugging in the above stress tensor gives:

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

\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:

\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 \beta satisfying

\beta^2 = \frac{\mu}{\rho}
This describes Swave propagation. Taking the divergence of seismic wave equation in homogeneous media, instead of the curl, yields an equation describing Pwave propagation. The steadystate SH waves are defined by the Helmholtz equation

(\nabla^2+k^2 )\boldsymbol{u}=0 ^{[2]}
where k is the wave number.
See also
References

^ University of Illinois at Chicago (17 July 1997). "Lecture 16 Seismographs and the earth's interior". Retrieved 8 June 2010.

^ Sheikhhassani, Ramtin (2013). "Scattering of a plane harmonic SH wave by multiple layered inclusions". Wave Motion 51 (3): 517–532.
Further reading

Shearer, Peter (1999). Introduction to Seismology (1st ed.). Cambridge University Press.


Fowler, C. M. R. (1990). The solid earth. Cambridge, UK: Cambridge University Press.
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