S-wave
Encyclopedia
A type of seismic wave
Seismic wave
Seismic waves are waves of energy that travel through the earth, and are a result of an earthquake, explosion, or a volcano that imparts low-frequency acoustic energy. Many other natural and anthropogenic sources create low amplitude waves commonly referred to as ambient vibrations. Seismic waves...

, the S-wave, secondary wave, or shear wave (sometimes called an elastic S-wave) is one of the two main types of elastic body waves, so named because they move through the body of an object, unlike surface wave
Surface wave
In physics, a surface wave is a mechanical wave that propagates along the interface between differing media, usually two fluids with different densities. A surface wave can also be an electromagnetic wave guided by a refractive index gradient...

s.

The S-wave moves as a shear or transverse wave
Transverse wave
A transverse wave is a moving wave that consists of oscillations occurring perpendicular to the direction of energy transfer...

, so motion is perpendicular to the direction of wave propagation: S-waves are like waves in a rope, as opposed to waves moving through a slinky
Slinky
Slinky or "Lazy Spring" is a toy consisting of a helical spring that stretches and can bounce up and down. It can perform a number of tricks, including traveling down a flight of steps end-over-end as it stretches and re-forms itself with the aid of gravity and its own momentum.-History:The toy was...

, the P-wave
P-wave
P-waves are a type of elastic wave, also called seismic waves, that can travel through gases , solids and liquids, including the Earth. P-waves are produced by earthquakes and recorded by seismographs...

. The wave moves through elastic media, and the main restoring force comes from shear effects. These waves do not diverge, and they obey the continuity equation for incompressible media:


Its name, S for secondary, comes from the fact that it is the second direct arrival on an earthquake seismogram
Seismogram
A seismogram is a graph output by a seismograph. It is a record of the ground motion at a measuring station as a function of time. Seismograms typically record motions in three cartesian axes , with the z axis perpendicular to the Earth's surface and the x- and y- axes parallel to the surface...

, after the compressional primary wave, or P-wave, because S-waves travel slower in rock. Unlike the P-wave, the S-wave cannot travel through the molten outer core
Outer core
The outer core of the Earth is a liquid layer about 2,266 kilometers thick composed of iron and nickel which lies above the Earth's solid inner core and below its mantle. Its outer boundary lies beneath the Earth's surface...

 of the Earth, and this causes a shadow zone
Shadow zone
A seismic shadow zone is an area of the Earth's surface where seismographs cannot detect an earthquake after its seismic waves have passed through the Earth. When an earthquake occurs, seismic waves radiate out spherically from the earthquake's focus...

 for S-waves opposite to where they originate. They can still appear in the solid inner core
Inner core
The inner core of the Earth, its innermost hottest part as detected by seismological studies, is a primarily solid ball about in radius, or about 70% that of the Moon...

: when a P-wave strikes the boundary of molten and solid cores, called the Lehmann discontinuity
Lehmann discontinuity
The Lehmann discontinuity refers to an abrupt increase of P-wave and S-wave velocities in the vicinity of 220±30 km depth, discovered by seismologist Inge Lehmann. It appears beneath continents, but not usually beneath oceans, and does not readily appear in globally averaged studies...

, S-waves will then propagate in the solid medium. And when the S-waves hit the boundary again they will in turn create P-waves. This property allows seismologists
Seismology
Seismology is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other planet-like bodies. The field also includes studies of earthquake effects, such as tsunamis as well as diverse seismic sources such as volcanic, tectonic, oceanic,...

 to determine the nature of the inner core.

As transverse waves, S-waves exhibit properties, such as polarization and birefringence
Birefringence
Birefringence, or double refraction, is the decomposition of a ray of light into two rays when it passes through certain anisotropic materials, such as crystals of calcite or boron nitride. The effect was first described by the Danish scientist Rasmus Bartholin in 1669, who saw it in calcite...

, much like other transverse waves. S-waves polarized in the horizontal plane are classified as SH-waves. If polarized in the vertical plane, they are classified as SV-waves. When an S- or P-wave strikes an interface at an angle other than 90 degrees, a phenomenon known as mode conversion occurs. As described above, if the interface is between a solid and liquid, S becomes P or vice versa. However, even if the interface is between two solid media, mode conversion results. If a P-wave strikes an interface, four propagation modes may result: reflected and transmitted P and reflected and transmitted SV. Similarly, if an SV-wave strikes an interface, the same four modes occur in different proportions. The exact amplitudes of all these waves are described by the Zoeppritz equations
Zoeppritz equations
In seismology, the Zoeppritz equations describe how seismic waves are transmitted and reflected at media boundaries, which are boundaries between two different layers of earth....

, which in turn are solutions to the wave equation
Wave equation
The wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...

. S-waves are slower than P-waves.

Theory

The prediction of S-waves came out of theory in the 1800s. Starting with the stress-strain
Deformation (mechanics)
Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration. A configuration is a set containing the positions of all particles of the body...

 relationship for an isotropic solid:


where is the stress, and are the Lamé parameters (with as the shear modulus), is the Kronecker delta, and the strain tensor is defined


for strain displacement u. Plugging the latter into the former yields:


Newton's 2nd law in this situation gives the homogeneous equation of motion for seismic wave propagation:


where is the mass density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

. Plugging in the above stress tensor gives:


Applying vector identities and making certain approximations gives the seismic wave equation in homogeneous media:


where Newton's notation has been used for the time derivative. Taking the curl of this equation and applying vector identities eventually gives:


which is simply the wave equation
Wave equation
The wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...

 applied to the curl of u with a velocity satisfying

This describes S-wave propagation. Taking the divergence
Divergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...

 of seismic wave equation in homogeneous media instead of the curl, yields an equation describing P-wave propgation.

See also

  • P-Wave
    P-wave
    P-waves are a type of elastic wave, also called seismic waves, that can travel through gases , solids and liquids, including the Earth. P-waves are produced by earthquakes and recorded by seismographs...

  • Love wave
    Love wave
    In elastodynamics, Love waves are horizontally polarized shear 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...

  • Rayleigh wave
    Rayleigh wave
    Rayleigh waves are a type of surface acoustic wave that travels on solids. They are produced on the Earth by earthquakes, in which case they are also known as "ground roll", or by other sources of seismic energy such as ocean waves an explosion or even a sledgehammer impact...

  • Lamb waves
    Lamb waves
    Lamb waves propagate in solid plates. They are elastic waves whose particle motion lies in the plane that contains the direction of wave propagation and the plate normal . In 1917, the English mathematician Horace Lamb published his classic analysis and description of acoustic waves of this type....

  • Longitudinal wave
    Longitudinal wave
    Longitudinal waves, as known as "l-waves", are waves that have the same direction of vibration as their direction of travel, which means that the movement of the medium is in the same direction as or the opposite direction to the motion of the wave. Mechanical longitudinal waves have been also...

  • Surface wave
    Surface wave
    In physics, a surface wave is a mechanical wave that propagates along the interface between differing media, usually two fluids with different densities. A surface wave can also be an electromagnetic wave guided by a refractive index gradient...

  • Shear wave splitting
    Shear wave splitting
    Shear wave splitting, also called seismic birefringence, is the phenomenon that occurs when a polarized shear wave enters an anisotropic media . The incident shear wave splits into two polarized shear waves . Shear wave splitting are typically used as a tool for testing the anisotropy of an area...

  • Earthquake Early Warning (Japan)
    Earthquake Early Warning (Japan)
    The is a warning issued just after an earthquake in Japan is detected. The warnings are issued mainly by the Japan Meteorological Agency , and they issue tips on how to react to the warnings.-Introduction:...

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