In solid mechanics, 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 direction of particle movement of an S wave is 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. Similarly, S waves cannot travel through gases. 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 solids. 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 a {\displaystyle a} and the other having a speed a 3 {\displaystyle {\frac {a}{\sqrt {3}}}} . 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 u = ( u 1 , u 2 , u 3 ) {\displaystyle {\boldsymbol {u}}=(u_{1},u_{2},u_{3})} be the displacement vector of a particle of such a medium from its "resting" position x = ( x 1 , x 2 , x 3 ) {\displaystyle {\boldsymbol {x}}=(x_{1},x_{2},x_{3})} due elastic vibrations, understood to be a function of the rest position x {\displaystyle {\boldsymbol {x}}} and time t {\displaystyle t} . The deformation of the medium at that point can be described by the strain tensor e {\displaystyle {\boldsymbol {e}}} , the 3×3 matrix whose elements are
e i j = 1 2 ( ∂ i u j + ∂ j u i ) {\displaystyle e_{ij}={\tfrac {1}{2}}\left(\partial _{i}u_{j}+\partial _{j}u_{i}\right)}
where ∂ i {\displaystyle \partial _{i}} denotes partial derivative with respect to position coordinate x i {\displaystyle x_{i}} . The strain tensor is related to the 3×3 stress tensor τ {\displaystyle {\boldsymbol {\tau }}} by the equation
τ i j = λ δ i j ∑ k e k k + 2 μ e i j {\displaystyle \tau _{ij}=\lambda \delta _{ij}\sum _{k}e_{kk}+2\mu e_{ij}}
Here δ i j {\displaystyle \delta _{ij}} is the Kronecker delta (1 if i = j {\displaystyle i=j} , 0 otherwise) and λ {\displaystyle \lambda } and μ {\displaystyle \mu } are the Lamé parameters ( μ {\displaystyle \mu } being the material's shear modulus). It follows that
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