Rayleigh wave
Rayleigh waves are a type of
Characteristics
Rayleigh waves are a type of surface wave that travel near the surface of solids. Rayleigh waves include both longitudinal and transverse motions that decrease exponentially in amplitude as distance from the surface increases. There is a phase difference between these component motions.[1]
The existence of Rayleigh waves was predicted in 1885 by
Rayleigh waves have a speed slightly less than shear waves by a factor dependent on the elastic constants of the material.[1] The typical speed of Rayleigh waves in metals is of the order of 2–5 km/s, and the typical Rayleigh speed in the ground is of the order of 50–300 m/s for shallow waves less than 100-m depth and 1.5–4 km/s at depths greater than 1 km. 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. This slow decay is one reason why they are of particular interest to seismologists. Rayleigh waves can circle the globe multiple times after a large earthquake and still be measurably large. There is a difference in the behavior (Rayleigh wave velocity, displacements, trajectories of the particle motion, stresses) of Rayleigh surface waves with positive and negative Poisson's ratio.[3]
In seismology, Rayleigh waves (called "ground roll") are the most important type of surface wave, and can be produced (apart from earthquakes), for example, by ocean waves, by explosions, by railway trains and ground vehicles, or by a sledgehammer impact.[1][4]
Speed and dispersion
In isotropic, linear elastic materials described by Lamé parameters and , Rayleigh waves have a speed given by solutions to the equation
where , , , and .[5] Since this equation has no inherent scale, the boundary value problem giving rise to Rayleigh waves are dispersionless. An interesting special case is the Poisson solid, for which , since this gives a frequency-independent phase velocity equal to . For linear elastic materials with positive Poisson ratio (), the Rayleigh wave speed can be approximated as , where is the shear-wave velocity.[6]
The elastic constants often change with depth, due to the changing properties of the material. This means that the velocity of a Rayleigh wave in practice becomes dependent on the
In non-destructive testing
Rayleigh waves are widely used for materials characterization, to discover the mechanical and structural properties of the object being tested – like the presence of cracking, and the related shear modulus. This is in common with other types of surface waves.[7] The Rayleigh waves used for this purpose are in the ultrasonic 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.
In electronic devices
Rayleigh waves propagating at high ultrasonic frequencies (10–1000 MHz) are used widely in different electronic devices.
In geophysics
Generation from earthquakes
Because Rayleigh waves are surface waves, the amplitude 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
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
- 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.
In seismology
Low frequency Rayleigh waves generated during
Possible animal reaction
Low frequency (< 20 Hz) Rayleigh waves are inaudible, yet they can be detected by many mammals, birds, insects and spiders. Humans 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.[10] Note that these Rayleigh waves have a much higher frequency than Rayleigh waves generated by earthquakes.
After the
See also
References
- ^ ISBN 978-0-521-33938-4. Retrieved 8 June 2011.
- ^ [1][dead link] "On Waves Propagated along the Plane Surface of an ElasticSolid", Lord Rayleigh, 1885
- S2CID 121607244.
- S2CID 31828394.
- ISBN 978-0-7506-2633-0.
- ISBN 978-0521629225.
- ISBN 978-0-306-45597-1. Retrieved 8 June 2011.
- ISBN 978-3540085751.
- ISBN 978-3-642-57767-3.
- PMID 11144599.
- ^ Kenneally, Christine (30 December 2004). "Surviving the Tsunami". www.slate.com. Retrieved 26 November 2013.
Further reading
- Viktorov, I.A. (2013) "Rayleigh and Lamb Waves: Physical Theory and Applications", Springer; Reprint of the original 1st 1967 edition by Plenum Press, New York. ISBN 978-1489956835.
- Aki, K. and Richards, P. G. (2002). Quantitative Seismology (2nd ed.). University Science Books. ISBN 0-935702-96-2.
- 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
- Sugawara, Y.; Wright, O. B.; Matsuda, O.; Takigahira, M.; Tanaka, Y.; Tamura, S.; Gusev, V. E. (18 April 2002). "Watching Ripples on Crystals". Physical Review Letters. 88 (18). American Physical Society (APS): 185504. PMID 12005696.