Asymptotic Analysis of the Modes of Wave Propagation in a Solid Cylinder

An asymptotic method is used to analyze the free vibrations of a solid elastic cylinder of circular cross section. In this method, the displacement components and the frequencies are sought as power series of the dimensionless wavenumber, where By substituting the expansions in the displacement equations of motion and the boundary conditions, and by collecting terms of the same order ⁿ, a system of coupled second order inhomogeneous ordinary differential equations is obtained with the radial variable as independent variable. Subsequent integration yields the coefficients of ⁿ for the displacement components and the frequencies for all modes and for the whole range of frequencies, but in a range of real valued dimensionless wavenumbers 0 & & 1.