Overturning of nonlinear compressional and shear waves subject to power-law attenuation or relaxation

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Despite accounting for certain loss mechanisms, model equations for nonlinear compressional and shear waves may predict waveform overturning (multivalued waveforms) when the source amplitude exceeds a critical value. Waveform overturning reveals that essential physics is not represented in the mathematical model. To restore physical relevance when overturning occurs, the model must be supplemented by weak-shock theory or augmented to include an additional loss mechanism. The present work determines the critical source amplitudes associated with two loss operators, one that accounts for power-law attenuation and the accompanying dispersion, and the other for relaxation. Radiation of a progressive plane wave from a monofrequency source is assumed. Both compressional and shear waves are considered, corresponding to quadratic and cubic nonlinearity, respectively. The model equations are transformed into intrinsic coordinates [Hammerton and Crighton, J. Fluid Mech 252, 585 (1993)], enabling numerical simulation of waveform evolution up to and beyond the distance at which the time derivative of the waveform first becomes infinite, denoting the onset of waveform overturning. It is found that attenuation coefficients that are proportional to frequency raised to a power less than unity, as well as the attenuation and dispersion corresponding to a single relaxation mechanism, are insufficient to prevent waveform overturning at all source amplitudes. In the parameter space for which overturning occurs, the distance at which an infinite derivative first appears in the time waveform is determined as a function of source amplitude.