Asymptotic study of dispersion relations for viscoelastic waveguides of canonical shapes in the case of Rabotnov's rheological model

Dispersion properties of a viscoelastic layer and a thin-walled hollow circular cylinder are studied by asymptotic methods. The viscoelastic properties of the material are described by the constitutive equations in the integral form with the fractional exponential Rabotnov's function as a kernel of the integral operators. For both types of waveguides, the generalized power series expansion are used to construct the asymptotic expansions of the lowest dispersion equations roots for small and large values of frequency. The influence of viscoelastic properties on the asymptotic behavior of dispersion curves is discussed. Comparison of the asymptotic solutions with the numerical one is presented.