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__NOTOC__ In
seismology Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other ...
and other areas involving elastic waves, S waves, secondary waves, or shear waves (sometimes called elastic S waves) are a type of
elastic wave 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 mec ...
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 wave In physics, a surface wave is a mechanical wave that propagates along the Interface (chemistry), interface between differing media. A common example is gravity waves along the surface of liquids, such as ocean waves. Gravity waves can also occu ...
s. S waves are
transverse wave In physics, a transverse wave is a wave whose oscillations are perpendicular to the direction of the wave's advance. This is in contrast to a longitudinal wave which travels in the direction of its oscillations. Water waves are an example of t ...
s, meaning that the direction of
particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object which can be described by several physical property, physical or chemical property, chemical ...
motion of a S wave is perpendicular to the direction of wave propagation, and the main restoring force comes from
shear stress Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ot ...
. Therefore, S waves cannot propagate in liquids with zero (or very low)
viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inte ...
; 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 A seismometer is an instrument that responds to ground noises and shaking such as caused by earthquakes, volcanic eruptions, and explosions. They are usually combined with a timing device and a recording device to form a seismograph. The output ...
, after the compressional primary wave, or
P wave A P wave (primary wave or pressure wave) is one of the two main types of elastic body waves, called seismic waves in seismology. P waves travel faster than other seismic waves and hence are the first signal from an earthquake to arrive at any ...
, because S waves travel more slowly in solids. Unlike P waves, S waves cannot travel through the molten
outer core Earth's outer core is a fluid layer about thick, composed of mostly iron and nickel that lies above Earth's solid inner core and below its mantle. The outer core begins approximately beneath Earth's surface at the core-mantle boundary and e ...
of the Earth, and this causes a
shadow zone A seismic shadow zone is an area of the Earth's surface where seismographs cannot detect direct P waves and/or S waves from an earthquake. This is due to liquid layers or structures within the Earth's surface. The most recognized shadow zone is ...
for S waves opposite to their origin. They can still propagate through the solid
inner core Earth's inner core is the innermost geologic layer of planet Earth. It is primarily a solid ball with a radius of about , which is about 20% of Earth's radius or 70% of the Moon's radius. There are no samples of Earth's core accessible for di ...
: 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 Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other ...
to determine some physical properties of the Earth's inner core.


History

In 1830, the mathematician
Siméon Denis Poisson Baron Siméon Denis Poisson FRS FRSE (; 21 June 1781 – 25 April 1840) was a French mathematician and physicist who worked on statistics, complex analysis, partial differential equations, the calculus of variations, analytical mechanics, electri ...
presented to the
French Academy of Sciences The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV of France, Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific me ...
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 \frac. At a sufficient distance from the source, when they can be considered
plane wave In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, th ...
s 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). 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'' / ''" ... (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 n the elastic solid of the Earthonly 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.)


Theory


Isotropic medium

For the purpose of this explanation, a solid medium is considered
isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
if its strain (deformation) in response to
stress Stress may refer to: Science and medicine * Stress (biology), an organism's response to a stressor such as an environmental condition * Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
is the same in all directions. Let \boldsymbol = (u_1,u_2,u_3) be the displacement
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
of a particle of such a medium from its "resting" position \boldsymbol=(x_1,x_2,x_3) due elastic vibrations, understood to be a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
of the rest position \boldsymbol and time t. The deformation of the medium at that point can be described by the
strain tensor In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally ...
\boldsymbol, the 3×3 matrix whose elements are e_ = \tfrac \left( \partial_i u_j + \partial_j u_i \right) where \partial_i denotes partial derivative with respect to position coordinate x_i. The strain tensor is related to the 3×3 stress tensor \boldsymbol by the equation \tau_ = \lambda\delta_\sum_ e_ + 2\mu e_ Here \delta_ is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
(1 if i = j, 0 otherwise) and \lambda and \mu are the
Lamé parameters In continuum mechanics, Lamé parameters (also called the Lamé coefficients, Lamé constants or Lamé moduli) are two material-dependent quantities denoted by λ and μ that arise in strain-stress relationships. In general, λ and μ are indi ...
(\mu being the material's
shear modulus In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain: :G \ \stackrel ...
). It follows that \tau_ = \lambda\delta_ \sum_ \partial_k u_k + \mu \left( \partial_i u_j + \partial_j u_i \right) From Newton's law of inertia, one also gets \rho \partial_t^2 u_i = \sum_j \partial_j\tau_ where \rho is the
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
(mass per unit volume) of the medium at that point, and \partial_t denotes partial derivative with respect to time. Combining the last two equations one gets the ''seismic wave equation in homogeneous media'' \rho \partial_t^2 u_i = \lambda\partial_i \sum_k \partial_k u_k + \mu\sum_j \bigl(\partial_i\partial_j u_j + \partial_j\partial_j u_i\bigr) Using the
nabla operator Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes t ...
notation of
vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...
, \nabla = (\partial_1, \partial_2, \partial_3), with some approximations, this equation can be written as \rho \partial_t^2 \boldsymbol = \left(\lambda + 2\mu \right) \nabla\left(\nabla \cdot \boldsymbol\right) - \mu\nabla \times \left(\nabla \times \boldsymbol\right) Taking the
curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was fi ...
of this equation and applying vector identities, one gets \partial_t^2(\nabla\times\boldsymbol) = \frac\nabla^2 \left(\nabla\times\boldsymbol\right) This formula is the
wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
applied to the vector quantity \nabla\times \boldsymbol, which is the material's shear strain. Its solutions, the S waves, are linear combinations of
sinusoidal A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in m ...
plane wave In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, th ...
s of various
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
s and directions of propagation, but all with the same speed \beta = \sqrt Taking the
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
of seismic wave equation in homogeneous media, instead of the curl, yields a wave equation describing propagation of the quantity \nabla \cdot \boldsymbol, which is the material's compression strain. The solutions of this equation, the P waves, travel at the speed \alpha = \sqrt that is more than twice the speed \beta of S waves. The
steady state In systems theory, a system or a Process theory, process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those p ...
SH waves are defined by the
Helmholtz equation In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation \nabla^2 f = -k^2 f, where is the Laplace operator (or "Laplacian"), is the eigenv ...
\left(\nabla^2 + k^2 \right) \boldsymbol=0 where is the wave number.


See also

* Earthquake Early Warning (Japan) *
Lamb waves Lamb waves propagate in solid plates or spheres. They are elastic waves whose particle motion lies in the plane that contains the direction of wave propagation and the direction perpendicular to the plate. In 1917, the English mathematician Horace ...
*
Longitudinal wave Longitudinal waves are waves in which the vibration of the medium is parallel ("along") to the direction the wave travels and displacement of the medium is in the same (or opposite) direction of the wave propagation. Mechanical longitudinal waves ...
*
Love wave 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 e ...
*
P wave A P wave (primary wave or pressure wave) is one of the two main types of elastic body waves, called seismic waves in seismology. P waves travel faster than other seismic waves and hence are the first signal from an earthquake to arrive at any ...
*
Rayleigh wave Rayleigh waves are a type of surface acoustic wave that travel along the surface of solids. They can be produced in materials in many ways, such as by a localized impact or by piezo-electric transduction, and are frequently used in non-destructiv ...
*
Seismic wave A seismic wave is a wave of acoustic energy that travels through the Earth. It can result from an earthquake, volcanic eruption, magma movement, a large landslide, and a large man-made explosion that produces low-frequency acoustic energy. S ...
*
Shear wave splitting Shear wave splitting, also called seismic birefringence, is the phenomenon that occurs when a Polarization (waves), polarized shear wave enters an anisotropic medium (Fig. 1). The incident shear wave splits into two polarized shear waves (Fig. 2). ...


References


Further reading

* * * {{Geotechnical engineering Waves Seismology