Causal time domain parabolic wave equations for power law absorption

Abstract

The classic parabolic time domain wave equation describes acoustic propagation in a medium in which absorption is a quadratic function of frequency. For the general case of power law absorption, α(ω)=α0‖ω‖y, where α0 and y are arbitrary real positive constants and ω is angular frequency, generalized time domain parabolic wave equations are presented. The differential loss operator in the original classic parabolic equation is replaced by a single propagation convolution operator that accounts for both absorption and dispersion. These operators, based on new time causal relations, have different forms for y as an even or odd integer or noninteger. The new equations are compared to those in the literature corresponding to the cases y=0.5 (acoustic duct), y=1.0 (medical and underwater applications), and y=0 or 2 (classic forms).