Evanescent waves are of increasing interest in certain applications such as the solution of scattering problems, but the research on the evanescent waves is quite limited and almost no detail has been analyzed regarding of imaginary and complex branches for all the modes for functionally graded structures. This paper presents an analytical approach, based on the orthogonal function technique, to solve the evanescent waves in a cylindrical curved plate with functionally graded material in the radial direction. Results are compared with the available data to check the validity of the present approach, and the convergence is also discussed. Dispersion characteristics of the circumferential guided waves in various graded cylindrical curved plates are investigated, and dispersion curves are plotted in tridimensional frequency-complex wavenumber space. The effects of different radius-thickness ratios and graded fields on the dispersion curves of the guided evanescent waves are illustrated. The displacement distributions are also discussed to analyze the specificities of the guided evanescent waves.