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In elastodynamics, Love waves, named after
Augustus Edward Hough Love Augustus Edward Hough Love FRS (17 April 1863, Weston-super-Mare – 5 June 1940, Oxford), often known as A. E. H. Love, was a mathematician famous for his work on the mathematical theory of elasticity. He also worked on wave propagation and h ...
, are horizontally polarized
surface wave In physics, a surface wave is a mechanical wave that propagates along the interface between differing media. A common example is gravity waves along the surface of liquids, such as ocean waves. Gravity waves can also occur within liquids, at th ...
s. The Love wave is a result of the
interference Interference is the act of interfering, invading, or poaching. Interference may also refer to: Communications * Interference (communication), anything which alters, modifies, or disrupts a message * Adjacent-channel interference, caused by extr ...
of many shear waves (
S-wave __NOTOC__ 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 ...
s) 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 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 ...
, Love waves (also known as Q waves (''Q''uer: German for lateral)) are
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
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 en ...
s that cause horizontal shifting of the Earth during an
earthquake An earthquake (also known as a quake, tremor or temblor) is the shaking of the surface of the Earth resulting from a sudden release of energy in the Earth's lithosphere that creates seismic waves. Earthquakes can range in intensity, fro ...
. Augustus Edward Hough Love predicted the existence of Love waves mathematically in 1911. They form a distinct class, different from other types of
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 en ...
s, such as
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 ...
s and
S-wave __NOTOC__ 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 ...
s (both body waves), or
Rayleigh waves 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 ...
(another type of surface wave). 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.


Description

The particle motion of a Love wave forms a horizontal line perpendicular to the direction of
propagation Propagation can refer to: *Chain propagation in a chemical reaction mechanism *Crack propagation, the growth of a crack during the fracture of materials *Propaganda, non-objective information used to further an agenda *Reproduction, and other forms ...
(i.e. are
transverse waves 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 ...
). Moving deeper into the material, motion can decrease to a "node" and then alternately increase and decrease as one examines deeper layers of particles. The
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
, or maximum particle motion, often decreases rapidly with depth. Since Love waves travel on the Earth's surface, the strength (or amplitude) of the waves decrease exponentially with the depth of an earthquake. However, given their confinement to the surface, their amplitude decays only as \frac, where r represents the distance the wave has travelled from the earthquake. Surface waves therefore decay more slowly with distance than do body waves, which travel in three dimensions. Large earthquakes may generate Love waves that travel around the Earth several times before dissipating. Since they decay so slowly, Love waves are the most destructive outside the immediate area of the focus or
epicentre The epicenter, epicentre () or epicentrum in seismology is the point on the Earth's surface directly above a hypocenter or focus, the point where an earthquake or an underground explosion originates. Surface damage Before the instrumental pe ...
of an earthquake. They are what most people feel directly during an earthquake. In the past, it was often thought that animals like cats and dogs could predict an earthquake before it happened. However, they are simply more sensitive to ground vibrations than humans and are able to detect the subtler body waves that precede Love waves, like the P-waves and the S-waves.


Basic theory

The conservation of
linear momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass a ...
of a linear elastic material can be written as The body force is assumed to be zero and direct tensor notation has been used. For other ways of writing these governing equations see
linear elasticity 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 ...
.
:\boldsymbol\cdot(\mathsf:\boldsymbol\mathbf) = \rho~\ddot where \mathbf is the
displacement vector In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along a s ...
and \mathsf is the
stiffness tensor Stiffness is the extent to which an object resists deformation in response to an applied force. The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is. Calculations The stiffness, k, of a bo ...
. Love waves are a special solution (\mathbf) that satisfy this system of equations. We typically use a Cartesian coordinate system (x,y,z) to describe Love waves. Consider an isotropic linear elastic medium in which the elastic properties are functions of only the z coordinate, i.e., 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 ind ...
and the mass
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. Mathematica ...
can be expressed as \lambda(z), \mu(z), \rho(z). Displacements (u,v,w) produced by Love waves as a function of time (t) have the form : u(x,y,z,t) = 0 ~,~~ v(x,y,z,t) = \hat(x,z,t) ~,~~ w(x,y,z,t) = 0 \,. These are therefore antiplane shear waves perpendicular to the (x,z) plane. The function \hat(x,z,t) can be expressed as the superposition of harmonic waves with varying
wave number In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
s (k) and
frequencies Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
(\omega). Consider a single harmonic wave, i.e., : \hat(x,z,t) = V(k, z, \omega)\,\exp (k x - \omega t) where i is the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition a ...
, i.e. i^2 = -1. The stresses caused by these displacements are : \sigma_ = 0 ~,~~ \sigma_ = 0 ~,~~ \sigma_ = 0 ~, ~~ \tau_ = 0 ~,~~ \tau_ = \mu(z)\,\frac\,\exp (k x - \omega t) ~,~~ \tau_ = i k \mu(z) V(k, z, \omega) \,\exp (k x - \omega t)\,. If we substitute the assumed displacements into the equations for the conservation of momentum, we get a simplified equation : \frac\left mu(z)\,\frac\right= ^2\,\mu(z) - \omega^2\,\rho(z),V(k,z,\omega) \,. The boundary conditions for a Love wave are that the surface tractions at the free surface (z = 0) must be zero. Another requirement is that the stress component \tau_ in a layer medium must be continuous at the interfaces of the layers. To convert the second order
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...
in V into two first order equations, we express this stress component in the form : \tau_ = T(k, z, \omega)\,\exp (k x - \omega t) to get the first order conservation of momentum equations : \frac\begin V \\ T \end = \begin 0 & 1/\mu(z) \\ k^2\,\mu(z) - \omega^2\,\rho(z) & 0 \end \begin V \\ T \end \,. The above equations describe an
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
problem whose solution
eigenfunctions In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, thi ...
can be found by a number of
numerical methods Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
. Another common, and powerful, approach is the propagator matrix method (also called the matricant approach).{{fact, date=February 2020


See also

*
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 w ...
* Antiplane shear


References

* A. E. H. Love, "Some problems of geodynamics", first published in 1911 by the Cambridge University Press and published again in 1967 by Dover, New York, USA. (Chapter 11: Theory of the propagation of seismic waves) Geophysics Surface waves Seismology