Polarization of Rayleigh Waves as a Function of Depth for Different Poisson's Ratio

In this work we study the polarization of Rayleigh waves in relation to depth for different Poisson's ratio. Rayleigh waves have the following features: the amplitude decreases rapidly with depth and the velocity is smaller than the velocity of body waves (P and S). Rayleigh waves are of great importance both in earthquake seismology and exploration seismology. Perhaps the best example is the presence of a Rayleigh wave called ground roll in the data collected in the field. In general, ground roll is considered noise in the seismic data, so that must be removed during the seismic processing stage, or at least have its effect attenuated. In the general theory of Rayleigh waves appears a cubic equation with a very special role. The roots of this cubic equation can be related to the Poisson's ratio ν . In general one assumes the condition of a Poisson solid, where ν = 0.25. Few researchers payed attention to other values of ν . In this work we calculated the velocity values of Rayleigh waves within a wide range of Poisson's ratio, from 0.00 to 0.50, where the interval is Δ ν = 0.01. The choice of this range is due to the fact that zero is the minimum theoretical value for Poisson's ratio and 0.50 is the maximum one. We show the relation between ν and three quantities: c/c₂, c/c₁, and c_{2/c_1}, where c₁ is the velocity of the P wave, c₂ is velocity of the S wave, and c is the velocity of Rayleigh wave. The first quantity may be denoted by γ = c/c₂. For each Poisson's ratio there are three roots, since the equation is cubic. For some values of Poisson's ratio although the velocity of the Rayleigh wave exist, that is, it is real, the surface wave itself does not exist. For each Poisson's ratio we calculated the path of particles confirming the elliptic behavior. The particle motion is retrograde when the displacements u and v are positive, and direct when u is negative and v is positive. In the case of Poisson solid (ν = 0.25), the real root of the cubic equation is γ ² = 0.8453, or γ = 0.9194, which means that the surface wave velocity c is 0.9194 times the value of S wave velocity c₂. It is this value generally mentioned in textbooks.