Numerical three-dimensional solution of broadband acoustic pulses through media with power-law attenuation

Abstract

Acoustic waves in tissues and weakly attenuative fluids often have an attenuation parameter, α, satisfying α=α0ωy, in which ω is the applied frequency and y is between 1 and 2. This power-law attenuation is not predicted by the classical thermoviscous wave equation, and recent research has led to a number of modified viscous wave equations in which the third term usually consists of a convolution operator or a fractional spatial or temporal derivative. These wave equations are obtained by taking into account the requisite wave velocity dispersion predicted by the attenuation in order for the signals to be causal. In this paper, acoustic waves undergoing power-law attenuation are modeled by a slight modification to the thermoviscous wave equation, in which the time derivative of the viscous term is replaced by a fractional time derivative. This new equation satisfies the power-law formulation for lossy waves. An explicit time-domain, finite-element formulation leads to a stable algorithm capable of simulating three-dimensional broadband acoustic pulses propagating through attenuative and dispersive media. Results are given for pulse propagation through layered lossy media, and it is shown how attenuation affects the transmission and reflection of broadband signals.