Rayleigh wave

Rayleigh waves are a type of surface acoustic wave that travel along the surface of solids. They can be produced in materials in many ways, such as by a localized impact or by piezo-electric transduction, and are frequently used in non-destructive testing for detecting defects. Rayleigh waves are part of the seismic waves that are produced on the Earth by earthquakes. When guided in layers they are referred to as Lamb waves, Rayleigh–Lamb waves, or generalized Rayleigh waves. Rayleigh waves are a type of surface acoustic wave that travel along the surface of solids. They can be produced in materials in many ways, such as by a localized impact or by piezo-electric transduction, and are frequently used in non-destructive testing for detecting defects. Rayleigh waves are part of the seismic waves that are produced on the Earth by earthquakes. When guided in layers they are referred to as Lamb waves, Rayleigh–Lamb waves, or generalized Rayleigh waves. 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. The existence of Rayleigh waves was predicted in 1885 by Lord Rayleigh, after whom they were named. In isotropic solids these waves cause the surface particles to 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 a particle is counterclockwise when the wave travels from left to right. At greater depths the particle motion becomes prograde. In addition, the motion amplitude decays and the eccentricity changes as the depth into the material increases. The depth of significant displacement in the solid is approximately equal to the acoustic wavelength. Rayleigh waves are distinct from other types of surface or guided acoustic waves such as Love waves or Lamb waves, both being types of guided waves supported by a layer, or longitudinal and shear waves, that travel in the bulk. Rayleigh waves have a speed slightly less than shear waves by a factor dependent on the elastic constants of the material. 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 linear elastic materials with positive Poisson ratio ( ν > 0 {displaystyle u >0} ), the Rayleigh wave speed can be approximated as c R / c S = 0.862 + 1.14 ν 1 + ν {displaystyle c_{R}/c_{S}={frac {0.862+1.14 u }{1+ u }}} . Since Rayleigh waves are confined near the surface, their in-plane amplitude when generated by a point source decays only as 1 / r {displaystyle {1}/{sqrt {r}}} , where r {displaystyle 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. 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. 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. In isotropic, linear elastic materials described by Lamé coefficients λ {displaystyle lambda } and μ {displaystyle mu } , Rayleigh waves have a speed given by solutions to the equation where ζ = ω 2 / k 2 β 2 {displaystyle zeta =omega ^{2}/k^{2}eta ^{2}} , η = β 2 / α 2 {displaystyle eta =eta ^{2}/alpha ^{2}} , ρ α 2 = λ + 2 μ {displaystyle ho alpha ^{2}=lambda +2mu } , and ρ β 2 = μ {displaystyle ho eta ^{2}=mu } . 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 λ = μ {displaystyle lambda =mu } , since this gives a frequency-independent phase velocity equal to ω / k = β 0.8453 {displaystyle omega /k=eta {sqrt {0.8453}}} . 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 wavelength (and therefore frequency), a phenomenon referred to as dispersion. Waves affected by dispersion have a different wave train shape. Rayleigh waves on ideal, homogeneous and flat elastic solids show no dispersion, as stated above. 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 multi-layered structures. 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. The Rayleigh waves used for this purpose are in the ultrasonic frequency range.