Wave hierarchies in continua with scalar microstructure in the plane and spherical symmetry

In this paper, we consider the balance equations for a continuum with scalar microstructure and propose a general approach to investigate nonlinear wave propagation by means of an asymptotic approach within the theoretical context of wave hierarchies. The evolution equations that are obtained for the leading terms of the asymptotics allow for the description of various levels of wave motion in agreement with the different scales involved in the modelling of continua with microstructure. Both the one-dimensional planar and the axisymmetric spherical cases are considered in a unified way, and various examples (immiscible mixtures of perfect fluids, granular materials, liquid with bubbles) are discussed in order to illustrate the procedure.