Development of asymptotic methods for the analysis of dispersion relations for a viscoelastic solid cylinder

Propagation of time-harmonic waves in a viscoelastic solid cylinder is considered. Vibrations of the cylinder are described by three-dimensional viscoelasticity equations in cylindrical coordinates. The stress-free surface boundary conditions are imposed. Viscoelastic properties are described by integral operators with a fractional-exponential kernel. For the case of a rational singularity parameter the method of asymptotic analysis of dispersion relations is proposed, which is based on the generalized power series expansion. For the axisymmetric waves the asymptotic expansions of the dispersion equation roots are obtained for low and high frequencies. The numerical results are presented to confirm the applicability of the proposed method.