Abstract: It is very difficult to solve the problem of the evanescent guided wave described by the complex root. To deal with this problem, an improved orthogonal polynomial method was proposed, which could obtain all the real, imaginary and complex solutions. For validation and illustration purposes, the proposed approach was applied to solving the problem of the dispersion curves of the evanescent guided waves in a cylindrical curved plate with a big radius-thickness ratio. The results agree well with the results of the plate from the existing literature. Dispersion curves of the circumferential evanescent guided waves in various cylindrical curved plates were obtained by using the present approach, and the three dimensional full dispersion curves were also obtained. The effects of different radius-thickness ratios and elastic parameters on the dispersion characteristics of circumferential evanescent guided waves in cylindrical curved plates were illustrated.