# Rayleigh waves: Wikis

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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 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.

## Characteristics

Picture of a Rayleigh wave.

Rayleigh waves travel across surfaces. In isotropic solids the surface particles move in ellipses in planes normal to the surface and parallel to the direction of propagation. At the surface and at shallow depths this motion is retrograde. Particles deeper in the material move in smaller ellipses with an eccentricity that changes with depth. 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 waves or Lamb waves, both being types of guided wave in a layer, or longitudinal and shear waves, that travel in the bulk.

Since Rayleigh waves are confined near the surface, their in-plane amplitude when generated by a point source decays only as ${1}/{\sqrt{r}}$, where r 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.

### Rayleigh wave dispersion

Dispersion of Rayleigh waves in a thin gold film on glass.[1]

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 seismograms recorded at distant earthquake recording stations. It is also possible to observe Rayleigh wave dispersion in thin films or multilayed structures.

## Rayleigh waves in non-destructive testing

Rayleigh waves are widely used for materials characterization 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.

Rayleigh waves in the ultrasonic frequency range are used in non-destructive testing applications to help find cracks and other imperfections in materials.

## Rayleigh waves in geophysics

### Rayleigh waves from earthquakes

Since Rayleigh waves are surface waves, the amplitude of such waves generated by an earthquake generally decreases exponentially with the depth of the hypocenter. 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-waves and S-waves, 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, 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:

• The size of the earthquake.
• The distance to the earthquake.
• The depth of the earthquake.
• The geologic structure of the crust.
• The focal mechanism 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 seismology to characterise the Earth's interior. In intermediate ranges, Rayleigh waves are used in geophysics and geotechnical engineering for the characterisation of soil deposits. These application 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.

## Other manifestations

### Animals

Low frequency (< 20 Hz) Rayleigh waves are inaudible, yet they can be detected by many mammals, birds, insects and spiders. Human beings should be able to detect such Rayleigh waves through their Pacinian corpuscles, 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 elephants may use vocalizations to generate Rayleigh waves. Since Rayleigh waves decay slowly, they should be detectable over long distances.[2] 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. [3]

• 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

• Dilbag Singh and S. K. Tomar, "Rayleigh–Lamb waves in a microstretch elastic plate cladded with liquid layers" Journal of Sound and Vibration, Volume 302, Issues 1-2, 17 April 2007, Pages 313-331