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In seismology and other areas involving elastic waves, 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. [1]
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. [2] Therefore, S waves cannot propagate in liquids [3] with zero (or very low) viscosity; however, they may propagate in liquids with high viscosity. [4] [5]
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. [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 and the other having a 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. 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
Here is the Kronecker delta (1 if , 0 otherwise) and and are the Lamé parameters ( being the material's shear modulus). It follows that
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
Using the nabla operator notation of vector calculus, , 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 , 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 . 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. Thus, .
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.
The steady state SH waves are defined by the Helmholtz equation [9]
where k is the wave number.
Similar to in an elastic medium, in a viscoelastic material, the speed of a shear wave is described by a similar relationship , however, here, 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: , where is the stiffness of the material and is the viscosity. [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 wave lengths 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. [10]
Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a continuous medium rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century.
The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves or electromagnetic waves. It arises in fields like acoustics, electromagnetism, and fluid dynamics.
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The Navier–Stokes equations are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics.
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In differential geometry, the Cotton tensor on a (pseudo)-Riemannian manifold of dimension n is a third-order tensor concomitant of the metric. The vanishing of the Cotton tensor for n = 3 is necessary and sufficient condition for the manifold to be locally conformally flat. By contrast, in dimensions n ≥ 4, the vanishing of the Cotton tensor is necessary but not sufficient for the metric to be conformally flat; instead, the corresponding necessary and sufficient condition in these higher dimensions is the vanishing of the Weyl tensor, while the Cotton tensor just becomes a constant times the divergence of the Weyl tensor. For n < 3 the Cotton tensor is identically zero. The concept is named after Émile Cotton.
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In differential geometry, the four-gradient is the four-vector analogue of the gradient from vector calculus.
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In fluid dynamics, Airy wave theory gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the fluid layer has a uniform mean depth, and that the fluid flow is inviscid, incompressible and irrotational. This theory was first published, in correct form, by George Biddell Airy in the 19th century.
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The acoustoelastic effect is how the sound velocities of an elastic material change if subjected to an initial static stress field. This is a non-linear effect of the constitutive relation between mechanical stress and finite strain in a material of continuous mass. In classical linear elasticity theory small deformations of most elastic materials can be described by a linear relation between the applied stress and the resulting strain. This relationship is commonly known as the generalised Hooke's law. The linear elastic theory involves second order elastic constants and yields constant longitudinal and shear sound velocities in an elastic material, not affected by an applied stress. The acoustoelastic effect on the other hand include higher order expansion of the constitutive relation between the applied stress and resulting strain, which yields longitudinal and shear sound velocities dependent of the stress state of the material. In the limit of an unstressed material the sound velocities of the linear elastic theory are reproduced.
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In optics, the Ewald–Oseen extinction theorem, sometimes referred to as just the extinction theorem, is a theorem that underlies the common understanding of scattering. It is named after Paul Peter Ewald and Carl Wilhelm Oseen, who proved the theorem in crystalline and isotropic media, respectively, in 1916 and 1915. Originally, the theorem applied to scattering by an isotropic dielectric objects in free space. The scope of the theorem was greatly extended to encompass a wide variety of bianisotropic media.
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S-wave.