Numerical model of a causal and fractional all-frequency wave equation for lossy media

Abstract

A numerical algorithm, to simulate a lossy acoustic wave equation with fractional time derivative terms, is presented. The inclusion of fractional derivatives yields a causal acoustic wave equation which can model and predict power law attenuation for which the level of absorption is proportional to frequency raised to a non-integer power. The fractional Zener wave equation is derived from the fractional Zener stress-strain constitutive relation and contains two absorption terms. One term which includes the Laplacian, similar to the traditional wave equation in viscous fluids, has a coefficient, which is proportional to a relaxation time. The other term is a fractional time derivative, higher than second order, and is proportional to the creep or retardation time. Both relaxation and retardation parameters will affect the frequency behavior of the attenuation. The inclusion of two terms enables the modeling of power-law attenuation in all frequency ranges. It results in more stable numerical simulations as both the overall mass and stiffness matrices of the finite element algorithm are changed according to the level of absorption. Results show the animation of diffraction patterns of focused and planar acoustic waves by inclusions with different values for the two different parameters, retardation time and relaxation time.