S wave

In seismology, S waves, secondary waves, or shear waves (sometimes called elastic S waves) 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.
S waves are transverse waves, meaning that the oscillations of an S wave's particles are perpendicular to the direction of wave propagation, and the main restoring force comes from shear stress. Therefore, S waves cannot propagate in liquids with zero (or very low) viscosity; however, they may propagate in liquids with high viscosity. 
The name "secondary wave" comes from the fact that they are the second type of wave to be detected by an earthquake seismograph, after the compressional primary wave, or P wave, because S waves travel more slowly in rock. Unlike P waves, S waves cannot travel through the molten outer core of the Earth, and this causes a shadow zone for S waves opposite to their origin. 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. This property allows seismologists to determine some physical properties of the Earth's inner core.
History.
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 formula_1 and the other having a speed formula_2. 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).
Theory.
Isotropic medium.
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 formula_3 be the displacement vector of a particle of such a medium from its "resting" position formula_4 due elastic vibrations, understood to be a function of the rest position formula_5 and time formula_6. The deformation of the medium at that point can be described by the strain tensor formula_7, the 3×3 matrix whose elements are
where formula_9 denotes partial derivative with respect to position coordinate formula_10. The strain tensor is related to the 3×3 stress tensor formula_11 by the equation
Here formula_13 is the Kronecker delta (1 if formula_14, 0 otherwise) and formula_15 and formula_16 are the Lamé parameters (formula_16 being the material's shear modulus). It follows that
From Newton's law of inertia, one also gets
where formula_20 is the density (mass per unit volume) of the medium at that point, and formula_21 denotes partial derivative with respect to time. Combining the last two equations one gets the "seismic wave equation in homogeneous media"
Using the nabla operator notation of vector calculus, formula_23, with some approximations, this equation can be written as
Taking the curl of this equation and applying vector identities, one gets
This formula is the wave equation applied to the vector quantity formula_26, 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 formula_27
Taking the divergence of seismic wave equation in homogeneous media, instead of the curl, yields a wave equation describing propagation of the quantity formula_28, which is the material's compression strain. The solutions of this equation, the P waves, travel at the speed formula_29 that is more than twice the speed formula_30 of S waves.
The steady-state SH waves are defined by the Helmholtz equation
where k is the wave number.