Comparative Analysis of Propagation of Own Damping Waves in A Viscoelastic Waveguide of a Sectoral Cross Section

Abstract

The main purpose of the article is a comparative study of the existence of natural waves in an infinite viscoelastic waveguide with a sectoral cross section and a plate (based on the hypotheses of Kirchhoff and Timoshenko) with a variable cross section (in the form of a wedge) depending on various parameters of the object (wave number and geometric parameters). Using the Navier equations and the physical relation, a system of six differential equations in partial derivatives for a sector waveguide is obtained. After simple transformations, a system of differential equations with complex coefficients is obtained, which is further solved using the method of straight lines, which will allow using the software tool of the orthogonal sweep method with a combination of the Muller method on complex arithmetic in solving. Also, on the basis of the variational principle, a system of ordinary differential equations with complex coefficients is given. On the basis of numerical calculations, it has been established that the real parts of the phase velocity of propagation of the first mode are less from the Rayleigh wave velocity to 20%, on the segment   (the central angle in the wedge-shaped waveguide), and further asymptotically approaches the Rayleigh wave, which propagates in the viscoelastic half-plane. Similar results were obtained for a wedge-shaped plate according to the theory of plates by Kirchhoff and Timoshenko, in the entire wave range. The results on the dynamic theory of elasticity and the plate (based on the hypotheses of Kirchhoff and Timoshenko) differ by no more than 6% for wedge apex angles not exceeding 28°. In     calculation results differ up to 20 %.  It has been established that at small wedge angles, the simplified theory of Kirchhoff – Lyava  and Timoshenko can be used in the entire wave range.