In this short paper we study unsteady motions of a new class of elastic solids, wherein one can justify a non-linear relationship between the linearized strain and the stress, an impossibility within the classical construct of elasticity. For the class of materials concerned, one has to solve simultaneously the balance of mass, balance of linear momentum and the constitutive relation. In general, one has ten scalar unknowns, i.e., density ρ, the components of Cauchy stress T and displacement u, and ten scalar algebraic-partial differential equations, the balance of mass (1), the balance of linear momentum (3) and the constitutive equation (6). The stress wave that is generated is quite distinct from what one observes within the context of the classical theory. © 2014 Elsevier B.V.