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.
The S-wave is a transverse wave, meaning that, in the simplest situation, the oscillations of the particles of the medium are perpendicular to the direction of wave propagation, and the main restoring force comes from shear stress. Therefore, S-waves can't propagate in liquids with zero (or very low) viscosity; However, it may propagate in liquids with high viscocity.
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.
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 a and the other having a speed a/√. 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).
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. Let 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 and time . The deformation of the medium at that point can be described by the strain tensor , the 3×3 matrix whose elements are
where denotes partial derivative with respect to position coordinate . The strain tensor is related to the 3×3 stress tensor by the equation
From Newton's law of inertia, one also gets
where is the density (mass per unit volume) of the medium at that point, and denotes partial derivative with respect to time. Combining the last two equations one gets the seismic wave equation in homogeneous media
Taking the curl of this equation and applying vector identities, one gets
This formula is the wave equation applied to the vector quantity , which is the material's shear strain. 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
Taking the divergence of seismic wave equation in homogeneous media, instead of the curl, yields a wave equation describing propagation of the quantity , which is the material's compression strain. The solutions of this equation, the P-waves, travel at the speed that is more than twice the speed of S-waves.
where k is the wave number.
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