Asymptotic analysis of the torsional modes of wave propagation in a piezoelectric solid circular cylinder of (622) class

An asymptotic method due to “Achenbach” is used to analyze the torsional modes of wave propagation in a solid circular cylinder of piezoelectric material of (622) crystal class. Information obtained in this method is useful for the frequency spectrum at long wavelengths. In this method, the field variables and the frequency are expressed as power series of the dimensionless wavenumber, $\text{? = 2π × R/W}$. Substituting these expansions in the field equations and the boundary conditions, a system of coupled second order inhomogeneous ordinary differential equations with the radial coordinate as independent variable is obtained by collecting the terms of same order $\text{?}{}^{\mathrm{m}}$. Integration of such systems of differential equations yield the various terms in the series expansions for the above modes and for the whole range of frequencies, when the real-valued dimensionless wavenumber less than unity ($\text{0<!--?<1</mtext-->}$). To test the correctness of the present scheme, the roots of the exact frequency equation are computed in double precision and the results thus obtained are compared with the results obtained in the present analysis. The results agree upto five decimal places.