S wave
[2] Therefore, S waves cannot propagate in liquids[3] with zero (or very low) viscosity; however, they may propagate in liquids with high viscosity.
They can still propagate through 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.
[6] In 1830, the mathematician Siméon Denis Poisson presented to the French 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
At a sufficient distance from the source, when they can be considered plane 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 considered isotropic if its strain (deformation) in response to stress is the same in all directions.
be the displacement vector of a particle of such a medium from its "resting" position
due elastic vibrations, understood to be a function of the rest position
The deformation of the medium at that point can be described by the strain tensor
denotes partial derivative with respect to position coordinate
is the density (mass per unit volume) of the medium at that point, and
denotes partial derivative with respect to time.
Using the nabla operator notation of vector calculus,
Taking the curl of this equation and applying vector identities, one gets
This formula is the wave equation applied to the vector quantity
Its solutions, the S waves, are linear combinations of sinusoidal plane waves of various wavelengths and directions of propagation, but all with the same speed
Assuming that the medium of propagation is linear, elastic, isotropic, and homogeneous, this equation can be rewritten as
[8] where ω is the angular frequency and k is the wavenumber.
Taking the divergence of seismic wave equation in homogeneous media, instead of the curl, yields a wave equation describing propagation of the quantity
The solutions of this equation, the P waves, travel at the faster speed
The steady state SH waves are defined by the Helmholtz equation[9]
Similar to in an elastic medium, in a viscoelastic material, the speed of a shear wave is described by a similar relationship
is a complex, frequency-dependent shear modulus and
is the frequency dependent phase velocity.
[8] One common approach to describing the shear modulus in viscoelastic materials is through the Voigt Model which states:
[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 and magnetic 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 the shear modulus.
MRE has seen use in studies of a variety of human tissues including liver, brain, and bone tissues.
