Dispersion curves are essential for analyzing and understanding the physical behaviors of guided waves propagating in structures. For circumferential waves in annular structures, the calculation of dispersion curves can be very time consuming. Furthermore, due to the nature of the mathematical functions involved in the curved structure geometry, the solutions can become unstable at high frequencies. In this paper three asymptotic approximation methods are introduced, which resolve the instability problem and speed up the calculations. Comparison of dispersion relations evaluated from the exact and asymptotic approximation methods are presented, showing errors to be typically less than 0.1&percent;, except for the very low frequency region. Results obtained using a two dimensional finite element technique are also presented for comparison. © 2003 American Institute of Physics