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

Without incorporation of weak-shock theory, the lossless Burgers equation predicts a multivalued waveform for nonlinear propagation beyond a certain distance. Inclusion of thermoviscous attenuation, which increases quadratically with frequency, prevents the occurrence of multivalued waveforms. The same is true for any attenuation law that is proportional to frequency raised to an exponent greater than unity. For exponents less than unity the situation is less clear. For example, when attenuation is constant with frequency (exponent equal zero) there is a critical value of the source amplitude below which a multivalued waveform is predicted and above which it is not. Prediction of multivalued waveforms indicates that the mathematical model is inadequate and must be either supplemented by weak-shock theory, or augmented to include an additional loss factor. To investigate the prediction of multivalued waveforms subject to power-law attenuation with exponents between zero and unity a Burgers equation with the loss term expressed as a fractional derivative is used [Prieur and Holm, J. Acoust. Soc. Am. 130, 1125–1132 (2011)]. 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. These solutions are used to determine the parameter space in which initially sinusoidal plane waves are predicted to evolve into multivalued waveforms for power-law attenuation with exponents less than unity.