Proper normal waves in a two-layer tube taking into account the rheological properties of materials

¹ Depatrment of Mathematics, Tashkent Institute of Chemical Technology, 32, Navoi str., Tashkent city, Uzbekistan
² Bukhara branch of Institute of Mathematics named after V.I. Romanovski, 11, M.Iqbol str., Bukhara city, Uzbekistan
³ Department of Natural and Exact Sciences, Bukhara State Pedagogical Institute, 16, K. Murtazayev str., Bukhara city, Uzbekistan
⁴ Depatrment of Mathematics, Navoi Mining and Technology University, 27, Galaba str., Navoi sity, Uzbekistan

^* Corresponding author: matlab1962@mail.ru

Abstract

The article considers the problem of propagation of proper normal waves in a two-layer pipe, paying to attention the rheological properties of the material. The main purpose of the work is to develop a methodology, an algorithm for investigation the problems of wave propagation in extended two-layer dissipatively inhomogeneous cylindrical structures. The main integro-differential equations in the article are obtained on the basis of methods of the theory of visco-elasticity. In the course of the study, the method of separation of variables, the freezing method, the Muller method and the Godunov orthogonal run method were used. In this paper, for structurally inhomogeneous extended cylindrical mechanical systems, the dependences of several modes (three and four) of the complex phase velocity (real and imaginary parts) on the wave number are comparatively estimated; the results of calculations are compared with experimental and theoretical data of other researchers, and the use of asymptotic and numerical methods for solving dispersion equations with complex output parameters is justified.