Asymptotic waves in viscoanelastic media with memory by the double-scale method

In this paper asymptotic dissipative waves in viscoanelastic media with shape and volumetric memory, previously studied by the author in a more classical way, are studied from the point of view of the double-scale method. Then, a physical interpretation is given of a new (fast) variable ξ, related to the family of hypersurfaces S , across which the solutions or/and some of their derivatives vary steeply, whereas their variation is slow along S and related to the old variables x α (called slow variables). The double-scale method is sketched. The thermodynamical model governing in the three-dimensional case the motion of the rheological media under consideration is presented and many results regarding the propagation of a particular solution into a uniform unperturbed state and the approximation of the first order of the wave front of this solution are given in full detail and reviewed by the double-scale method. Other novel results are worked out. The thermodynamic models for viscoanelastic media with memory are applied in rheology and in several technological sectors.