Generalized viscoelastic wave equation

Abstract

This paper presents a generalized wave equation which unifies viscoelastic and pure elastic cases into a single wave equation. In the generalized wave equation, the degree of viscoelasticity varies between zero and unity, and is defined by a controlling parameter. When this viscoelastic controlling parameter equals to 0, the viscous property vanishes and the generalized wave equation becomes a pure elastic wave equation. When this viscoelastic controlling parameter equals to 1, it is the Stokes equation made up of a stack of pure elastic and Newtonian viscous models. Given this generalized wave equation, an analytical solution is derived explicitly in terms of the attenuation and the velocity dispersion. It is proved that, for any given value of the viscoelastic controlling parameter, the attenuation component of this generalized wave equation perfectly satisfies the power laws of frequency. Since the power laws are the fundamental characteristics in physical observations, this generalized wave equation can well represent seismic wave propagation through subsurface media.