Rayleigh wave

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

In isotropic, linear elastic materials described by Lamé parameters ${\displaystyle \lambda }$ and ${\displaystyle \mu }$, Rayleigh waves have a speed given by solutions to the equation

${\displaystyle \zeta ^{3}-8\zeta ^{2}+8\zeta (3-2\eta )-16(1-\eta )=0,}$

where ${\displaystyle \zeta =\omega ^{2}/k^{2}\beta ^{2}}$, ${\displaystyle \eta =\beta ^{2}/\alpha ^{2}}$, ${\displaystyle \rho \alpha ^{2}=\lambda +2\mu }$, and ${\displaystyle \rho \beta ^{2}=\mu }$.[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 ${\displaystyle \lambda =\mu }$, since this gives a frequency-independent phase velocity equal to ${\displaystyle \omega /k=\beta {\sqrt {0.8453}}}$. For linear elastic materials with positive Poisson ratio (${\displaystyle \nu >0.3}$), the Rayleigh wave speed can be approximated as ${\displaystyle c_{R}=c_{S}{\frac {0.862+1.14\nu }{1+\nu }}}$, where ${\displaystyle c_{S}}$ 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 wavelength (and therefore frequency), a phenomenon referred to as dispersion. Waves affected by dispersion have a different wave train shape.[1] 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.

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 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 (0.340 km/s), 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:

Rayleigh wave direction

• 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.

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 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.