In
elastodynamics
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 ...
, Love waves, named after
Augustus Edward Hough Love, are horizontally
polarized 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. 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 extra ...
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
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 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 ...
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, from ...
. 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 energy. S ...
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-waves (both
body waves), or
Rayleigh waves (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). 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 amplit ...
, 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
, where
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 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 Ground vibrations is a technical term that is being used to describe mostly man-made vibrations of the ground, in contrast to natural vibrations of the Earth studied by seismology. For example, vibrations caused by explosions, construction works, ra ...
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 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.]
:
where
is the
displacement vector and
is the
stiffness tensor. Love waves are a special solution (
) that satisfy this system of equations. We typically use a Cartesian coordinate system (
) to describe Love waves.
Consider an isotropic linear elastic medium in which the elastic properties are functions of only the
coordinate, i.e., the
Lamé parameters 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. Mathematical ...
can be expressed as
. Displacements
produced by Love waves as a function of time (
) have the form
:
These are therefore
antiplane shear
Antiplane shear or antiplane strainW. S. Slaughter, 2002, ''The Linearized Theory of Elasticity'', Birkhauser is a special state of strain in a body. This state of strain is achieved when the displacements in the body are zero in the plane of inte ...
waves perpendicular to the
plane. The function
can be expressed as the superposition of
harmonic wave
A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...
s with varying
wave numbers (
) 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 ...
(
). Consider a single harmonic wave, i.e.,
:
where
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 an ...
, i.e.
. The
stresses caused by these displacements are
:
If we substitute the assumed displacements into the equations for the conservation of momentum, we get a simplified equation
:
The boundary conditions for a Love wave are that the
surface traction
Traction, or tractive force, is the force used to generate motion between a body and a tangential surface, through the use of dry friction, though the use of shear force of the surface is also commonly used.
Traction can also refer to the ''maxim ...
s at the free surface
must be zero. Another requirement is that the stress component
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, an ...
in
into two first order equations, we express this stress component in the form
:
to get the first order conservation of momentum equations
:
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 b ...
problem whose solution
eigenfunctions 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
*
Antiplane shear
Antiplane shear or antiplane strainW. S. Slaughter, 2002, ''The Linearized Theory of Elasticity'', Birkhauser is a special state of strain in a body. This state of strain is achieved when the displacements in the body are zero in the plane of inte ...
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