Wave propagation in marine sediments expressed by fractional wave and diffusion equations

Abstract

Attenuation of compressional and shear waves in sediments often follows power laws with near linear variation with frequency. This cannot be modeled with viscous or relaxation wave equations, but more general temporal memory operators in the wave equation can describe such behavior. These operators can be justified in four ways: 1) Power laws for attenuation with exponents other than two correspond to the use of convolution operators with a kernel which is a power law in time. 2) The corresponding constitutive equation is also a convolution, often with a temporal power law function. 3) It is also equivalent to an infinite set of relaxation processes which can be formulated via the complex compressibility. 4) The constitutive equation can also be expressed as an infinite sum of higher order derivatives. We also analyze a grain-shearing model for propagation of waves in saturated, unconsolidated granular materials. It is expressed via a spring damper model with time-varying damping. It turns out that it results in a fractional Kelvin-Voigt wave equation and a fractional diffusion equation for the compressional and shear waves respectively, giving a new perspective for understanding and interpreting this model.