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

• Published in: Wave motion Vol. 85; pp. 18 - 33
• Main Authors: Cormack, John M., Hamilton, Mark F.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.01.2019; Elsevier BV
• Subjects: Accounting; Amplitudes; Attenuation coefficients; Computer simulation; Mathematical models; Nonlinear acoustics; Nonlinear equations; Nonlinearity; Operators (mathematics); Plane waves; Power law; Power-law attenuation; Propagation; Radiation; Relaxation; S waves; Shear; Wave attenuation; Waveform analysis; Waveform overturning; Waveforms; Relaxation; Nonlinear acoustics; Waveform overturning; Power-law attenuation
• ISSN: 0165-2125; 1878-433X
• DOI: 10.1016/j.wavemoti.2018.10.004

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. •Not all loss operators prevent nonlinear evolution of multivalued waveforms.•Multivalued waveforms indicate that a mathematical model lacks essential physics.•Maximum source amplitudes are determined for power-law attenuation and relaxation.•Results for nonlinear compressional and shear waves are qualitatively alike.•Analysis is based on formulation of mathematical models in intrinsic coordinates.