Burgers’ equation generalized for absorption obeying a power law

Abstract

Burgers’ equation applies to finite-amplitude waves propagating in a medium with absorption that has a quadratic frequency dependence. Numerical solutions of a modified Burgers’ equation have been obtained in the frequency domain for other types of losses; however, a complete set of time domain nonlinear equations corresponding to power law attenuation has not been available. Power law attenuation is defined by the equation, α(ω)=α0‖ω‖y, where α0 and y are arbitrary real constants, and ω is angular frequency. Blackstock [J. Acoust. Soc. Am. 77, 2050–2053 (1985)] has suggested that Burgers’ equation could be generalized if an appropriate operator L could be found such that the equation could become pz−L*p=Bppτ, where p is pressure, B is a constant, τ is delayed time, and the subscripts denote derivative operations. The L operators have been derived based on a new causality principle and parabolic wave equations for power law loss. The resulting time domain equations extend Burgers’ approach to finite amplitude propagation in media with arbitrary power law absorption.