An asymptotic averaged model of non-linear long waves propagation in media with a regular structure

An asymptotic averaged model describing long non-linear wave propagation in non-equilibrium structured media is suggested. The averaged system of the hydrodynamic equations is an integro-differential system which cannot be reduced to the system of averaged hydrodynamic values. It is proved that the long wave with finite amplitude responds to the structure of the medium so that the structured medium cannot be modeled as a homogeneous medium. However, for long acoustic waves the internal structure of a medium manifests itself only by means of the dispersive dissipative properties, and the dynamic behavior of the medium can be described in the framework of a homogeneous relaxing medium. The important result of this model is that the structure of a medium always increases the non-linear effects under the propagation of long waves, and that non-linearity takes place even for media with the components described by the linear law. On a micro structural level of a medium, the dynamic behavior adheres only to the thermodynamic laws, wherein the change of the structure eventually affects the macro wave motion. The model justifies the one-velocity continuous models. As an example, a comparison of the suggested model with the known Lyakhov's model for natural multicomponent media is carried out.