Rayleigh wave
Encyclopedia
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. They can also be produced in materials by many mechanisms, including by piezo-electric
transducers
, and are frequently used in non-destructive testing for detecting defects. When guided in layers they are referred to as Lamb waves
, Rayleigh–Lamb waves, or generalized Rayleigh waves.
Rayleigh waves travel across surfaces - and are thus a type of surface wave
. In perfectly homogenous, isotropic and infinite materials, Rayleigh waves would not be apparent. In seismology, Rayleigh waves (called "ground roll") are the most important type of surface wave. Most obvious close to the surface of mediums, Rayleigh waves are made of longitudinal and traverse motion that decreases exponentially in amplitude as distance from the surface increases. There is a distinct phase difference between the component motion types. In isotropic solids the surface particles move in ellipses in planes normal to the surface and parallel to the direction of propagation – the major axis of the ellipse is vertical. At the surface and at shallow depths this motion is retrograde, that is the in-plane motion of the wave at the peak is in the opposite direction to the propagation of the wave. Deeper into the material the motion amplitude decays and the eccentricity changes. At greater depths the particle motion becomes prograde. The depth of significant displacement in the solid is approximately equal to the acoustic wavelength
. Rayleigh waves are distinct from other types of acoustic
waves such as Love wave
s or Lamb waves, both being types of guided wave in a layer, or longitudinal and shear waves, that travel in the bulk.
Rayleigh waves have a speed less than S-waves, by a factor dependent on the elastic constant (σ) of the material near the surface.
Since Rayleigh waves are confined near the surface, their in-plane amplitude when generated by a point source decays only as
, where 
is the radial distance. Surface waves therefore decay more slowly with distance than do bulk waves, which spread out in three dimensions from a point source. The speed of Rayleigh waves on bulk solids, of the order of 2–5 km/s, is slightly less than the shear velocity
.
The existence of Rayleigh waves was predicted in 1885 by Lord Rayleigh, after whom they were named.
]
The elastic constant often changes with depth, due to the properties of the material. This means the velocity of a Rayleigh wave is dependent on the wavelength
(and therefore frequency
). Waves affected dispersion have a different wave train shape. Rayleigh waves on ideal, homogeneous and flat elastic solids show no dispersion. However, if a solid or structure has a density
or sound velocity that varies with depth, Rayleigh waves become dispersive. One example is Rayleigh waves on the Earth's surface: those waves with a higher frequency
travel more slowly than those with a lower frequency
. This occurs because a Rayleigh wave of lower frequency has a relatively long wavelength
. The displacement of long wavelength waves penetrates more deeply into the Earth than short wavelength waves. Since the speed of waves in the Earth increases with increasing depth, the longer wavelength (low frequency) waves can travel faster than the shorter wavelength (high frequency) waves. Rayleigh waves thus often appear spread out on seismogram
s recorded at distant earthquake recording stations. It is also possible to observe Rayleigh wave dispersion in thin films or multi-layered structures.
frequency range.
They are used at different length scales because they are easily generated and detected on the free surface of solid objects. Since they are confined in the vicinity of the free surface within a depth (~ the wavelength) linked to the frequency
of the wave, different frequencies can be used for characterization at different length scales.
of such waves generated by an earthquake generally decreases exponentially with the depth of the hypocenter
(focus). However, large earthquakes may generate Rayleigh waves that travel around the Earth several times before dissipating.
In seismology longitudinal and shear waves are known as P-wave
s and S-wave
s, respectively, and are termed body waves. Rayleigh waves are generated by the interaction of P- and S- waves at the surface of the earth, and travel with a velocity that is lower than the P-, S-, and Love wave velocities. Rayleigh waves emanating outward from the epicenter of an earthquake travel along the surface of the earth at about 10 times the speed of sound
(0.340 km/s), in air, that is ~3 km/s.
Due to their higher speed, the P- and S-waves generated by an earthquake arrive before the surface waves. However, the particle motion of surface waves is larger than that of body waves, so the surface waves tend to cause more damage. In the case of Rayleigh waves, the motion is of a rolling nature, similar to an ocean surface wave
. The intensity of Rayleigh wave shaking at a particular location is dependent on several factors:
Local geologic structure can serve to focus or defocus Rayleigh waves, leading to significant differences in shaking over short distances.
to characterise the Earth
's interior.
In intermediate ranges, Rayleigh waves are used in geophysics
and geotechnical engineering
for the characterisation of oil
deposits. These applications are based on the geometric dispersion
of Rayleigh waves and on the solution of an inverse problem on the basis of seismic data collected on the ground surface using active sources (falling weights, hammers or small explosions, for example) or by recording microtremors.
Rayleigh ground waves are important also for environmental noise and vibration control since they make a major contribution to traffic-induced ground vibrations
and the associated structure-borne noise
in buildings.
s, bird
s, insect
s and spider
s. Human beings should be able to detect such Rayleigh waves through their Pacinian corpuscle
s, which are in the joints, although people do not seem to consciously respond to the signals. Some animals seem to use Rayleigh waves to communicate. In particular, some biologists theorize that elephant
s may use vocalizations to generate Rayleigh waves. Since Rayleigh waves decay slowly, they should be detectable over long distances.http://biox.stanford.edu/grant/klemperer_r2_iip.html Note that these Rayleigh waves have a much higher frequency than Rayleigh waves generated by earthquakes.
After the 2004 Indian Ocean Earthquake
, some people have speculated that Rayleigh waves served as a warning to animals to seek higher ground, allowing them to escape the more slowly-traveling tsunami
. At this time, evidence for this is mostly anecdotal. Another animal early warning systems may rely on an ability to sense infrasonic
waves traveling through the air. http://www.slate.com/id/2111608
Surface acoustic wave
]A surface acoustic wave is an acoustic wave traveling along the surface of a material exhibiting elasticity, with an amplitude that typically decays exponentially with depth into the substrate.-Discovery:...
that travels on solids. They are produced on the Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
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. They can also be produced in materials by many mechanisms, including by piezo-electric
Piezoelectricity
Piezoelectricity is the charge which accumulates in certain solid materials in response to applied mechanical stress. The word piezoelectricity means electricity resulting from pressure...
transducers
Interdigital transducer
An interdigital transducer , or interdigitated transducer, is a device which consists of two interlocking comb-shaped metallic coatings which are applied to a piezoelectric substrate, such as quartz or lithium niobate. IDTs are primarily used to convert microwaves to surface acoustic waves ....
, and are frequently used in non-destructive testing for detecting defects. When guided in layers they are referred to as 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....
, Rayleigh–Lamb waves, or generalized Rayleigh waves.
Characteristics
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...
. In perfectly homogenous, isotropic and infinite materials, Rayleigh waves would not be apparent. In seismology, Rayleigh waves (called "ground roll") are the most important type of surface wave. Most obvious close to the surface of mediums, Rayleigh waves are made of longitudinal and traverse motion that decreases exponentially in amplitude as distance from the surface increases. There is a distinct phase difference between the component motion types. In isotropic solids the surface particles move in ellipses in planes normal to the surface and parallel to the direction of propagation – the major axis of the ellipse is vertical. At the surface and at shallow depths this motion is retrograde, that is the in-plane motion of the wave at the peak is in the opposite direction to the propagation of the wave. Deeper into the material the motion amplitude decays and the eccentricity changes. At greater depths the particle motion becomes prograde. The depth of significant displacement in the solid is approximately equal to the acoustic wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
. Rayleigh waves are distinct from other types of acoustic
Acoustics
Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics...
waves such as 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...
s or Lamb waves, both being types of guided wave in a layer, or longitudinal and shear waves, that travel in the bulk.
Rayleigh waves have a speed less than S-waves, by a factor dependent on the elastic constant (σ) of the material near the surface.
Since Rayleigh waves are confined near the surface, their in-plane amplitude when generated by a point source decays only as
, where is the radial distance. Surface waves therefore decay more slowly with distance than do bulk waves, which spread out in three dimensions from a point source. The speed of Rayleigh waves on bulk solids, of the order of 2–5 km/s, is slightly less than the shear velocityShear velocity
Shear velocity, also called friction velocity, is a form by which a shear stress may be re-written in units of velocity. It is useful as a method in fluid mechanics to compare true velocities, such as the velocity of a flow in a stream, to a velocity that relates shear between layers of flow.Shear...
.
The existence of Rayleigh waves was predicted in 1885 by Lord Rayleigh, after whom they were named.
Rayleigh wave dispersion
The elastic constant often changes with depth, due to the properties of the material. This means the velocity of a Rayleigh wave is dependent on the wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
(and therefore frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
). Waves affected dispersion have a different wave train shape. Rayleigh waves on ideal, homogeneous and flat elastic solids show no dispersion. However, if a solid or structure has a 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...
or sound velocity that varies with depth, Rayleigh waves become dispersive. One example is Rayleigh waves on the Earth's surface: those waves with a higher frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
travel more slowly than those with a lower frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
. This occurs because a Rayleigh wave of lower frequency has a relatively long wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
. The displacement of long wavelength waves penetrates more deeply into the Earth than short wavelength waves. Since the speed of waves in the Earth increases with increasing depth, the longer wavelength (low frequency) waves can travel faster than the shorter wavelength (high frequency) waves. Rayleigh waves thus often appear spread out on 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...
s recorded at distant earthquake recording stations. It is also possible to observe Rayleigh wave dispersion in thin films or multi-layered structures.
Rayleigh waves in non-destructive testing
Rayleigh waves are widely used for materials characterization, to discover the mechanical and structural properties of the object being testing – like the presence of cracking, and the related shear modulus. This is in common with other types of surface waves. The Rayleigh waves used for this purpose are in the ultrasonicUltrasound
Ultrasound is cyclic sound pressure with a frequency greater than the upper limit of human hearing. Ultrasound is thus not separated from "normal" sound based on differences in physical properties, only the fact that humans cannot hear it. Although this limit varies from person to person, it is...
frequency range.
They are used at different length scales because they are easily generated and detected on the free surface of solid objects. Since they are confined in the vicinity of the free surface within a depth (~ the wavelength) linked to the frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
of the wave, different frequencies can be used for characterization at different length scales.
Rayleigh waves from earthquakes
Since Rayleigh waves are surface waves, the amplitudeAmplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
of such waves generated by an earthquake generally decreases exponentially with the depth of the hypocenter
Hypocenter
The hypocenter refers to the site of an earthquake or a nuclear explosion...
(focus). However, large earthquakes may generate Rayleigh waves that travel around the Earth several times before dissipating.
In seismology longitudinal and shear waves are known as 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...
s and S-wave
S-wave
A type of seismic wave, the S-wave, secondary wave, or shear 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 waves....
s, respectively, and are termed body waves. Rayleigh waves are generated by the interaction of P- and S- waves at the surface of the earth, and travel with a velocity that is lower than the P-, S-, and Love wave velocities. Rayleigh waves emanating outward from the epicenter of an earthquake travel along the surface of the earth at about 10 times the speed of sound
Speed of sound
The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....
(0.340 km/s), in air, that is ~3 km/s.
Due to their higher speed, the P- and S-waves generated by an earthquake arrive before the surface waves. However, the particle motion of surface waves is larger than that of body waves, so the surface waves tend to cause more damage. In the case of Rayleigh waves, the motion is of a rolling nature, similar to an ocean surface wave
Ocean surface wave
In fluid dynamics, wind waves or, more precisely, wind-generated waves are surface waves that occur on the free surface of oceans, seas, lakes, rivers, and canals or even on small puddles and ponds. They usually result from the wind blowing over a vast enough stretch of fluid surface. Waves in the...
. The intensity of Rayleigh wave shaking at a particular location is dependent on several factors:
- The size of the earthquake.
- The distance to the earthquake.
- The depth of the earthquake.
- The geologic structure of the crust.
- The focal mechanismFocal mechanismThe focal mechanism of an earthquake describes the inelastic deformation in the source region that generates the seismic waves. In the case of a fault-related event it refers to the orientation of the fault plane that slipped and the slip vector and is also known as a fault-plane solution...
of the earthquake. - The rupture directivity of the earthquake.
Local geologic structure can serve to focus or defocus Rayleigh waves, leading to significant differences in shaking over short distances.
Rayleigh waves in seismology
Low frequency Rayleigh waves generated during earthquakes are used in seismologySeismology
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 characterise the Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
's interior.
In intermediate ranges, Rayleigh waves are used in geophysics
Geophysics
Geophysics is the physics of the Earth and its environment in space; also the study of the Earth using quantitative physical methods. The term geophysics sometimes refers only to the geological applications: Earth's shape; its gravitational and magnetic fields; its internal structure and...
and geotechnical engineering
Geotechnical engineering
Geotechnical engineering is the branch of civil engineering concerned with the engineering behavior of earth materials. Geotechnical engineering is important in civil engineering, but is also used by military, mining, petroleum, or any other engineering concerned with construction on or in the ground...
for the characterisation of oil
Oil
An oil is any substance that is liquid at ambient temperatures and does not mix with water but may mix with other oils and organic solvents. This general definition includes vegetable oils, volatile essential oils, petrochemical oils, and synthetic oils....
deposits. These applications are based on the geometric dispersion
Dispersion
Dispersion may refer to:In physics:*The dependence of wave velocity on frequency or wavelength:**Dispersion , for light waves**Dispersion **Acoustic dispersion, for sound waves...
of Rayleigh waves and on the solution of an inverse problem on the basis of seismic data collected on the ground surface using active sources (falling weights, hammers or small explosions, for example) or by recording microtremors.
Rayleigh ground waves are important also for environmental noise and vibration control since they make a major contribution to traffic-induced ground vibrations
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, railway and road transport, etc - all belong to...
and the associated structure-borne noise
Noise
In common use, the word noise means any unwanted sound. In both analog and digital electronics, noise is random unwanted perturbation to a wanted signal; it is called noise as a generalisation of the acoustic noise heard when listening to a weak radio transmission with significant electrical noise...
in buildings.
Animals
Low frequency (< 20 Hz) Rayleigh waves are inaudible, yet they can be detected by many mammalMammal
Mammals are members of a class of air-breathing vertebrate animals characterised by the possession of endothermy, hair, three middle ear bones, and mammary glands functional in mothers with young...
s, bird
Bird
Birds are feathered, winged, bipedal, endothermic , egg-laying, vertebrate animals. Around 10,000 living species and 188 families makes them the most speciose class of tetrapod vertebrates. They inhabit ecosystems across the globe, from the Arctic to the Antarctic. Extant birds range in size from...
s, insect
Insect
Insects are a class of living creatures within the arthropods that have a chitinous exoskeleton, a three-part body , three pairs of jointed legs, compound eyes, and two antennae...
s and spider
Spider
Spiders are air-breathing arthropods that have eight legs, and chelicerae with fangs that inject venom. They are the largest order of arachnids and rank seventh in total species diversity among all other groups of organisms...
s. Human beings should be able to detect such Rayleigh waves through their Pacinian corpuscle
Pacinian corpuscle
Lamellar corpuscles or Pacinian corpuscles are one of the four major types of mechanoreceptor. They are nerve endings in the skin, responsible for sensitivity to vibration and pressure. Vibrational role may be used to detect surface, e.g., rough vs...
s, which are in the joints, although people do not seem to consciously respond to the signals. Some animals seem to use Rayleigh waves to communicate. In particular, some biologists theorize that elephant
Elephant
Elephants are large land mammals in two extant genera of the family Elephantidae: Elephas and Loxodonta, with the third genus Mammuthus extinct...
s may use vocalizations to generate Rayleigh waves. Since Rayleigh waves decay slowly, they should be detectable over long distances.http://biox.stanford.edu/grant/klemperer_r2_iip.html Note that these Rayleigh waves have a much higher frequency than Rayleigh waves generated by earthquakes.
After the 2004 Indian Ocean Earthquake
2004 Indian Ocean earthquake
The 2004 Indian Ocean earthquake was an undersea megathrust earthquake that occurred at 00:58:53 UTC on Sunday, December 26, 2004, with an epicentre off the west coast of Sumatra, Indonesia. The quake itself is known by the scientific community as the Sumatra-Andaman earthquake...
, some people have speculated that Rayleigh waves served as a warning to animals to seek higher ground, allowing them to escape the more slowly-traveling tsunami
Tsunami
A tsunami is a series of water waves caused by the displacement of a large volume of a body of water, typically an ocean or a large lake...
. At this time, evidence for this is mostly anecdotal. Another animal early warning systems may rely on an ability to sense infrasonic
Infrasound
Infrasound is sound that is lower in frequency than 20 Hz or cycles per second, the "normal" limit of human hearing. Hearing becomes gradually less sensitive as frequency decreases, so for humans to perceive infrasound, the sound pressure must be sufficiently high...
waves traveling through the air. http://www.slate.com/id/2111608
See also
- Surface acoustic waveSurface acoustic wave]A surface acoustic wave is an acoustic wave traveling along the surface of a material exhibiting elasticity, with an amplitude that typically decays exponentially with depth into the substrate.-Discovery:...
- SeismologySeismologySeismology 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,...
- Longitudinal waveLongitudinal waveLongitudinal 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...
- Shear wave
- S-waveS-waveA type of seismic wave, the S-wave, secondary wave, or shear 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 waves....
- P wave
- Love waveLove waveIn 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...
- Linear elasticityLinear elasticityLinear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of...
- PhononPhononIn physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...
Further reading
- Viktorov, I.A. (1967) “Rayleigh and Lamb Waves: physical theory and applications”, Plenum Press, New York
- Aki, K. and Richards, P. 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.
- Lai, C.G., Wilmanski, K. (Eds.) (2005). Surface Waves in Geomechanics: Direct and Inverse Modelling for Soils and Rocks" Series: CISM International Centre for Mechanical Sciences , Number 481 , Springer, Wien, ISBN 978-3-211-27740-9
- Y. Sugawara, O. B. Wright, O. Matsuda, M. Takigahira, Y. Tanaka, S. Tamura and V. E. Gusev, "Watching ripples on crystals", Phys. Rev. Lett. 88, 185504 (2002)