The authors present the mathematical analysis for the calculation of the dispersion relation, the field distributions, and the ohmic losses for TEm,p modes in an infinite coaxial waveguide with a longitudinally corrugated insert. The method employed is based on an appropriate eigenfunction expansion, and its main advantage is the very fast convergence with a few spatial harmonics. The analysis is properly extended to include tapered cavities with varying, in respect to the z-coordinate, outer and/or inner radius. Numerical results are presented for several tapered cavity geometries and compared with already published methods