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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.^{ [1] }

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.^{ [2] } Therefore, S-waves can't 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 a earthquake seismogram, after the compressional primary wave, or P-wave, because S-waves travel slower 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^{[ inconsistent ]} 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

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 ^{ [8] }

where k is the wave number.

**Acoustic theory** is a scientific field that relates to the description of sound waves. It derives from fluid dynamics. See acoustics for the engineering approach.

In physics, the **Navier–Stokes equations**, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous fluid substances.

**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.

In fluid dynamics, the **Euler equations** are a set of quasilinear hyperbolic equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. The equations represent Cauchy equations of conservation of mass (continuity), and balance of momentum and energy, and can be seen as particular Navier–Stokes equations with zero viscosity and zero thermal conductivity. In fact, Euler equations can be obtained by linearization of some more precise continuity equations like Navier–Stokes equations in a local equilibrium state given by a Maxwellian. The Euler equations can be applied to incompressible and to compressible flow – assuming the flow velocity is a solenoidal field, or using another appropriate energy equation respectively. Historically, only the incompressible equations have been derived by Euler. However, fluid dynamics literature often refers to the full set – including the energy equation – of the more general compressible equations together as "the Euler equations".

In differential geometry, the **Cotton tensor** on a (pseudo)-Riemannian manifold of dimension *n* is a third-order tensor concomitant of the metric, like the Weyl tensor. The vanishing of the Cotton tensor for *n* = 3 is necessary and sufficient condition for the manifold to be conformally flat, as with the Weyl tensor for *n* ≥ 4. For *n* < 3 the Cotton tensor is identically zero. The concept is named after Émile Cotton.

In elastodynamics, **Love waves**, named after Augustus Edward Hough Love, are horizontally polarized surface waves. The Love wave is a result of the interference of many shear waves (S-waves) guided by an elastic layer, which is *welded* to an elastic half space on one side while bordering a vacuum on the other side. In seismology, **Love waves** are surface seismic waves that cause horizontal shifting of the Earth during an earthquake. Augustus Edward Hough Love predicted the existence of Love waves mathematically in 1911. They form a distinct class, different from other types of seismic waves, such as P-waves and S-waves, or Rayleigh waves. Love waves travel with a lower velocity than P- or S- waves, but faster than Rayleigh waves. These waves are observed only when there is a low velocity layer overlying a high velocity layer/ sub–layers.

In continuum mechanics, the **finite strain theory**—also called **large strain theory**, or **large deformation theory**—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strain theory. In this case, the undeformed and deformed configurations of the continuum are significantly different, requiring a clear distinction between them. This is commonly the case with elastomers, plastically-deforming materials and other fluids and biological soft tissue.

The **Maxwell stress tensor** is a symmetric second-order tensor used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum. In simple situations, such as a point charge moving freely in a homogeneous magnetic field, it is easy to calculate the forces on the charge from the Lorentz force law. When the situation becomes more complicated, this ordinary procedure can become impossibly difficult, with equations spanning multiple lines. It is therefore convenient to collect many of these terms in the Maxwell stress tensor, and to use tensor arithmetic to find the answer to the problem at hand.

A **hyperelastic** or **Green elastic** material is a type of constitutive model for ideally elastic material for which the stress–strain relationship derives from a strain energy density function. The hyperelastic material is a special case of a Cauchy elastic material.

The **Newman–Penrose** (**NP**) **formalism** is a set of notation developed by Ezra T. Newman and Roger Penrose for general relativity (GR). Their notation is an effort to treat general relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR. The NP formalism is itself a special case of the tetrad formalism, where the tensors of the theory are projected onto a complete vector basis at each point in spacetime. Usually this vector basis is chosen to reflect some symmetry of the spacetime, leading to simplified expressions for physical observables. In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a complex-conjugate pair. The two real members asymptotically point radially inward and radially outward, and the formalism is well adapted to treatment of the propagation of radiation in curved spacetime. The Weyl scalars, derived from the Weyl tensor, are often used. In particular, it can be shown that one of these scalars— in the appropriate frame—encodes the outgoing gravitational radiation of an asymptotically flat system.

**Nonlinear acoustics** (NLA) is a branch of physics and acoustics dealing with sound waves of sufficiently large amplitudes. Large amplitudes require using full systems of governing equations of fluid dynamics and elasticity. These equations are generally nonlinear, and their traditional linearization is no longer possible. The solutions of these equations show that, due to the effects of nonlinearity, sound waves are being distorted as they travel.

There are various **mathematical descriptions of the electromagnetic field** that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.

The intent of this article is to highlight the important points of the **derivation of the Navier–Stokes equations** as well as its application and formulation for different families of fluids.

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.

In continuum mechanics, a **compatible** deformation **tensor field** in a body is that *unique* tensor field that is obtained when the body is subjected to a continuous, single-valued, displacement field. **Compatibility** is the study of the conditions under which such a displacement field can be guaranteed. Compatibility conditions are particular cases of integrability conditions and were first derived for linear elasticity by Barré de Saint-Venant in 1864 and proved rigorously by Beltrami in 1886.

**Riemann invariants** are mathematical transformations made on a system of conservation equations to make them more easily solvable. Riemann invariants are constant along the characteristic curves of the partial differential equations where they obtain the name invariant. They were first obtained by Bernhard Riemann in his work on plane waves in gas dynamics.

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.

In general relativity, light is assumed to propagate in a vacuum along a null geodesic in a pseudo-Riemannian manifold. Besides the geodesics principle in a classical field theory there exists Fermat's principle for stationary gravity fields.

Plasticity theory for rocks is concerned with the response of rocks to loads beyond the elastic limit. Historically, conventional wisdom has it that rock is brittle and fails by fracture while plasticity is identified with ductile materials. In field scale rock masses, structural discontinuities exist in the rock indicating that failure has taken place. Since the rock has not fallen apart, contrary to expectation of brittle behavior, clearly elasticity theory is not the last work.

The **Batchelor–Chandrasekhar equation** is the evolution equation for the scalar functions, defining the two-point velocity correlation tensor of a homogeneous axisymmetric turbulence, named after George Batchelor and Subrahmanyan Chandrasekhar. They developed the theory of homogeneous axisymmetric turbulence based on Howard P. Robertson's work on isotropic turbulence using an invariant principle. This equation is an extension of Kármán–Howarth equation from isotropic to axisymmetric turbulence.

- ↑ What are seismic waves? UPSeis at Michigan Tech
- ↑ S wave US Geological Survey
- ↑ "Why can't S-waves travel through liquids?".
*Earth Observatory of Singapore*. Retrieved 2019-12-06. - ↑ Greenwood, Margaret Stautberg; Bamberger, Judith Ann (August 2002). "Measurement of viscosity and shear wave velocity of a liquid or slurry for on-line process control".
*Ultrasonics*.**39**(9): 623–630. doi:10.1016/s0041-624x(02)00372-4. ISSN 0041-624X. PMID 12206629. - ↑ "Do viscous fluids support shear waves propagation?".
*ResearchGate*. Retrieved 2019-12-06. - ↑ University of Illinois at Chicago (17 July 1997). "Lecture 16 Seismographs and the earth's interior". Archived from the original on 7 May 2002. Retrieved 8 June 2010.
- ↑ Poisson, S. D. (1831). "Mémoire sur la propagation du mouvement dans les milieux élastiques" [Memoir on the propagation of motion in elastic media].
*Mémoires de l'Académie des Sciences de l'Institut de France*(in French).**10**: 549–605. From p.595: "*On verra aisément que cet ébranlement donnera naissance à deux ondes sphériques qui se propageront uniformément, l'une avec une vitesse*a*, l'autre avec une vitesse*b*ou*a*/ √3*" … (One will easily see that this quake will give birth to two spherical waves that will be propagated uniformly, one with a speed*a*, the other with a speed*b*or*a*/√3 … ) From p.602: … "*à une grande distance de l'ébranlement primitif, et lorsque les ondes mobiles sont devenues sensiblement planes dans chaque partie très-petite par rapport à leurs surfaces entières, il ne subsiste plus que des vitesses propres des molécules, normales ou parallèles à ces surfaces ; les vitesses normal ayant lieu dans les ondes de la première espèce, où elles sont accompagnées de dilations qui leur sont proportionnelles, et les vitesses parallèles appartenant aux ondes de la seconde espèce, où elles ne sont accompagnées d'aucune dilatation ou condensation de volume, mais seulement de dilatations et de condensations linéaires.*" ( … at a great distance from the original quake, and when the moving waves have become roughly planes in every tiny part in relation to their entire surface, there remain [in the elastic solid of the Earth] only the molecules' own speeds, normal or parallel to these surfaces ; the normal speeds occur in waves of the first type, where they are accompanied by expansions that are proportional to them, and the parallel speeds belonging to waves of the second type, where they are not accompanied by any expansion or contraction of volume, but only by linear stretchings and squeezings.) - ↑ Sheikhhassani, Ramtin (2013). "Scattering of a plane harmonic SH wave by multiple layered inclusions".
*Wave Motion*.**51**(3): 517–532. doi:10.1016/j.wavemoti.2013.12.002.

- Shearer, Peter (1999).
*Introduction to Seismology*(1st ed.). Cambridge University Press. ISBN 0-521-66023-8. - Aki, Keiiti; Richards, Paul G. (2002).
*Quantitative Seismology*(2nd ed.). University Science Books. ISBN 0-935702-96-2. - Fowler, C. M. R. (1990).
*The solid earth*. Cambridge, UK: Cambridge University Press. ISBN 0-521-38590-3.S-wave.

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