SOLUTIONS OF THE WAVE EQUATION IN PROPROPAGATION OF MECHANICAL WAVES PROBLEM IN A TWO-DIMENSIONAL LAYERED MEDIUM WITH ALMOST INVARIANT RELATION OF MATERIAL AND GEOMETRIC PARAMETERS

The issues raised in the current work are oriented towards the prospective creation of developments in the field of realization of artificial properties in elastic internally structured (discontinuous) media in connection with the scattering of mechanical (acoustic) waves in them. The well-known principle of emergence predicts the appearance of properties in a system (in any system of arbitrary nature) that are not inherent to each constituent element separately. This principle encourages us to look at the realization of properties in these things precisely through the realization of structuredness. In particular, we are talking about elastic media in connection with the propagation of mechanical waves in them. The authors develop the opinion that a layered strictly periodic medium can be modified by material and geometric parameters (a flat model of the medium unlimited along the periodicity is considered). The premise for the development of this idea was the assumption of the existence of invariant forms in the dispersion equation (equations that connect the parameters of the environment with the conditions for the solvability of the Sturm-Liouville problem). For the implementation of these modifications, the apparatus of quasi-periodic functions is involved. Without going beyond the scope of generality, such an apparatus, in particular, allows you to formulate research results in terms inaccessible to the concepts of strictly periodic functions (almostperiod, almost-root, almost equal, etc.). The authors of the paper resort to attempts to solve the problem on its eigen oscillations (the Sturm-Liouville problem), putting forward the idea of ow to achieve the constituent condition of its solvability and, accordingly, write down the equation connecting this condition with the parameters of the environment (dispersion equation).