Modeling of plane progressive waves in media that exhibit slow dynamics and dissipative nonlinearity

Nonclassical nonlinearity refers to amplitude-dependent effects that are irreversible, for example slow dynamics and dissipative nonlinearity, both phenomena that are commonly observed in materials such as sandstone and concrete. Analyses of progressive waves in such media have been few, and have relied on hysteresis models that are difficult to implement, especially for transient or strongly nonlinear pulses. In this work, an evolution equation for plane progressive waves in a nonclassically nonlinear material is derived using two phenomenological models: the internal variable model of Berjamin et al. for slow dynamics [Proc. Roy. Soc. A 473, 20170024 (2017)], and a simple model for dissipative nonlinearity [Zaitsev and Nazarov, Acoust. Phys. 44, 362 (1998)]. The evolution equation is easily solved numerically given an arbitrary initial waveform. An initially narrowband signal exhibits pulse lengthening, amplitude-dependent attenuation, and nonclassical waveform steepening during propagation. Experimental observation of the same phenomena were reported recently by Remillieux et al. [J. Geophys. Res. 122, 8892 (2017)]. Their results are reproduced in this study using the new evolution equation, showing good agreement between the model and the measurements, with additional insight into the behavior of the material gained from the model. [J.M.C. supported by the ARL:UT McKinney Fellowship in Acoustics.]