Shear waves in a nonlinear relaxing media: A three-dimensional perspective

An important one-dimensional rheological model for the propagation of a linearly polarized shear wave was recently obtained and proposed by Cormack and Hamilton [(2018). J. Acoust. Soc. Am. 143(2), 1035-1048]. We show that it is possible to embed such a result within a wider and complete set of general three-dimensional models derived within the theoretical framework of rigorous continuum mechanics. We show that, following this approach, we are able to derive in a simple and straightforward way the equations that govern the propagation of circularly polarized shear waves. When the phase of such waves is constant, we find the same equation for linearly polarized shear waves already proposed elsewhere. Moreover, we show that, under appropriate asymptotic assumptions, our results are indifferent with respect to the choice of the objective time derivative used in the constitutive class.