A perturbation method is presented to analytically calculate eigensolutions of the two-dimensional wave equation when asymmetric perturbations are present in the boundary conditions. The unique feature of the method is that the sequence of boundary value problems governing the eigensolution perturbations are solved exactly through fifth order perturbation. Two classes of asymmetry are considered: irregular domain shapes that cannot be treated by analytical means, and variation of the boundary conditions along the boundary. The unperturbed eigensolutions are those for an annular domain with axisymmetric boundary conditions. Irregularly shaped domains are studied in detail to demonstrate the method and the accuracy of the results, which are compared with exact values for the elliptical and rectangular domain cases. The results show excellent agreement with these known solutions for large shape distortions, an achievement resulting from the extension to higher order perturbation. Fourier representation of the boundary asymmetries allows analysis of arbitrary distributions of asymmetry. Additionally, the exact perturbation solution retains the explicit parameter dependence of continuous system analysis, generates simple expressions for the perturbed eigensolutions, addresses all distinct and degenerate axisymmetric system eigensolutions, and requires minimal computation and programming.