PROPAGATION OF MECHANIC WAVE IN TWO-DIMENSION STRATIFIED MEDIUM

Abstract

The principle of emergence encourages us to look for new properties in future materials as a result of structuring of constituent elements, and mostly we are talking about non-obvious structuring. For example, understanding and simulating isotropy is on the way to robustness against internal vibrations. Structurality itself is a characteristic feature of the world around a person (protein molecules, crystal lattices, etc.). However, the variety of natural structures is so numerous that directly studying them to identify new properties appears to be quite a difficult task. Therefore, to a certain extent, the principle of emergence supposedly contrasts the need to copy or imitate natural structures with the ability to carry out inventions. Just as a wide variety of patterns are formed in an optical kaleidoscope thanks to 2 or 3 optical elements, the principle of emergence gives reason to expect that as a result of structuring only a few components of the future environment, new properties can be obtained. The development of technology, in particular, 3D printing, opens up opportunities to look at the methodology of structuring the environment in a new way. If, in addition, this process is relatively economic, then empirically, by carrying out structuring, it is possible to obtain experimental laboratory samples. Today they are talking about the possibility of achieving such effects as lensing of mechanical waves precisely thanks to emergent properties. The corresponding devices promise to find application, for example, in ultrasound diagnostics in medicine (Assoc. Prof. J. Memoli, University of Sussex, UK). The paper considers the problem of the propagation of mechanical waves in a folding periodic two-layer medium - a flat model. For such a medium, a wave equation is written and solved by the method of separation of variables. Such an equation turns out to be a linear differential equation with periodic piecewise-stable coefficients. From the basic theory of differential equations with periodic coefficients (Floquet theory), the method of transfer matrices (a method for calculating the passage of waves through multilayer media) is well known, which makes it possible to obtain a composite condition for the solvability of the wave equation. The work develops an approach to obtaining solvability conditions (constructing a dispersion equation) along with the classical method of transfer matrices. The proposed approach is in a certain sense equivalent to the transfer matrix method, however, it has some remarkable differences, in particular, it provides a rigorous mathematical basis for the transition to media with a finite number of layers.