PROPAGATION OF SHEAR WAVES IN VISCOELASTIC TWO- LAYER MEDIA Safarov Ismoil Ibrokhimovich, Doctor of physical-mechanical sciences, professor, professor of department of “Higher mathematics” of Tashkent Institute of Chemical Technology Teshaev Muhsin Khudoyberdiyevich, Doctor of physical-mechanical sciences, associated professor, chief researcher of the Bukhara branch of the Institute of Mathematics named after V.I. Romanovsky Otazhonova Nilufar Baxtiyorovna, Senior Lecturer of department of “Higher mathematics” of Tashkent State Pedagogical University Axmedov Maxsud Sharipovich, Doctor of philosophy in physical-mechanical sciences, Senior Lecturer of department of “Higher mathematics” of Bukhara Institute of Engineering and Technology Zhuraev Uktam Shavkatovich Doctor of philosophy in technical sciences, Senior Lecturer of department of “Construction” of Fergana Polytechnic Institute Abstract : The problem of propagation of shear waves in a viscoelastic
inhomogeneous two-layer medium is considered, when one layer is exponentially
inhomogeneous, in particular, a homogeneous viscoelastic can be. The main purpose
of the work is to develop a methodology and algorithm for solving the problem
of wave propagation in layered media. For the mathematical formulation of the
problem, the Lame equation and the corresponding boundary conditions are used.
The obtained dispersion equation describing the relationship between frequency
and wavenumber is investigated. The equation of dispersion relations is solved
numerically by the Muller method. For dissipative inhomogeneous mechanical
systems, it is found that there is a finite number of waves if the velocity of the
shear wave propagation is greater than the velocity of the bulk wave of the first
layer. It is also established that there are no shear waves propagating at a speed less
than the minimum velocity of the volume wave of the first layer. For a viscoelastic
material, a Pohhammer-type frequency equation is obtained and the dispersion of the
complex phase velocity is determined. The viscoelastic properties of the material are
described using the Rzhanitsyn–Koltunov inheritance kernel. Damping properties of
the structure are evaluated.
Keywords: wave, layer, free boundary, contact conditions, dispersion
relations
