Rayleigh Waves: Velocity, Attenuation with Depth and Elliptical Polarization

Abstract

In 1887, Lord Rayleigh published a theory of a surface wave that later became known as the Rayleigh wave. The main features of this wave are its attenuation with depth and the fact that its velocity is always less than the S-wave velocity. Classical textbooks on elastic waves propagation and seismology usually consider the so-called Poisson solid, in which the Poisson’s ratio is equal to 0.25. In this study, we present some mathematical details and geometrical aspects that are not discussed in the textbooks. We obtain the Rayleigh cubic equation and compute its three roots as a function of the Poisson’s ratio. Then, we compute the particle displacements in a given depth, U(z) and W(z), as well as the instantaneous displacements, u(x, z, t) and w(x, z, t), to demonstrate the elliptical polarization of the particle motion and to indicate the rotation direction of the geometrical locus. We find that for each value of Poisson’s ratio, at a critical depth, the rotation direction changes from retrograde to prograde. Finally, we find the critical depth by three different approaches.

Keywords

• Rayleigh wave
• Poisson’s ratio
• cubic equation
• ellipse rotation direction
• ellipse eccentricity