Nonlinear wave propagation in media with attenuation and dispersion modeled using a superposition of relaxation mechanisms

Abstract

Michael Buckingham has long held interest in modeling wave propagation in media with nontraditional attenuation and dispersion, especially unconsolidated marine sediments exhibiting power-law attenuation [Buckingham, J. Acoust. Soc. Am. 138, 2871 (2015)]. The present authors have examined the implications of power-law attenuation and single relaxation mechanisms for modeling nonlinear propagation of compressional and shear waves [Cormack and Hamilton, Wave Motion 85, 13 (2019)]. Reported here are results for media that are modeled using multiple relaxation mechanisms. Below the lowest relaxation frequency, attenuation increases as frequency squared, and above the highest relaxation frequency, attenuation is constant. In between, the superposition may be tailored to approximate power-law attenuation that increases in proportion to frequency raised to an exponent between 0 and 2. Regardless of the source frequency and the number of relaxation mechanisms incorporated, there exists a critical source amplitude above which the mathematical model is incapable of offsetting nonlinear waveform distortion sufficiently to stabilize shock formation beyond the distance where an infinite gradient first appears in the waveform. Results are presented for birelaxing media corresponding to seawater and atmosphere, followed by models based on multiple relaxation mechanisms that approximate power-law attenuation over limited frequency ranges in sediment and tissue.