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A type of elastic wave, the Swave, secondary wave, or shear wave (sometimes called an elastic Swave) is 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:
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, called the Lehmann discontinuity, 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.
The prediction of Swaves came out of theory in the 1800s. Starting with the stressstrain relationship for an isotropic solid in Einstein notation:
where is the stress, and are the Lamé parameters (with as the shear modulus), is the Kronecker delta, and the strain tensor is defined
for strain displacement u. Plugging the latter into the former yields:
Newton's 2nd law in this situation gives the homogeneous equation of motion for seismic wave propagation:
where is the mass density. Plugging in the above stress tensor gives:
Applying vector identities and making certain approximations gives the seismic wave equation in homogeneous media:
where Newton's notation has been used for the time derivative. Taking the curl of this equation and applying vector identities eventually gives:
which is simply the wave equation applied to the curl of u with a velocity satisfying
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
where k is the wave number.
