Discrete and continuous cases for the problem of propagating waves for inhomogeneous medium with memory

The article is devoted to the study of the wave equation for medium with memory. This equation is obtained in the process of considering the homogenized models of combined mediums. It describes one-dimensional case of the Kelvin–Voight's viscoelastic oscillations law of homogenized models. The problem is to find the function which describes the average offset of the material. The formula of propagating waves is used for this purpose. It allows to construct a solution using the general solution of the first order system in which each equation is the equation of the transfer along the corresponding characteristics. The main result consists of two theorems for discrete and continuous modification of the equation. Furthermore the article contains descriptive considerations which lead to the construction of the classical solution of the equations.