Overturning of nonlinear acoustic waves in media with power-law attenuation

Abstract

In the absence of losses, the equations of motion for nonlinear acoustic propagation of a progressive plane wave lead inevitably to the prediction of a multivalued waveform. Losses described by an attenuation coefficient that varies as frequency raised to a power greater than unity are sufficient to ensure single-valued waveforms in the presence of nonlinearity, regardless of amplitude. The situation is less clear for power-law attenuation with exponents less than unity. A Burgers equation with the loss term expressed as a fractional derivative [Prieur and Holm, J. Acoust. Soc. Am. 130, 1125-1132 (2011)] is used to investigate the prediction of multivalued waveforms subject to power-law attenuation with exponents between zero and unity. Transformation of the equation into intrinsic coordinates following Hammerton and Crighton [J.Fluid Mech. 252, 585-599 (1993)] permits simulation of waveforms beyond the point at which they become multivalued. The simulations are used to determine the parameter space in which initially sinusoidal plane waves are predicted to evolve into multivalued waveforms for powerlaw attenuation with exponents less than unity. Prediction of a multivalued waveform indicates that the mathematical model is inadequate and must be either supplemented by weak-shock theory or augmented to include an additional loss factor.