Higher-Order Terms in a Perturbation Expansion for the Impedance of a Corrugated Wave Guide

We consider a circular waveguide whose radius a(z) = a(1+εs(z)) varies periodically with the axial coordinate z. Chatard-Moulin and Papiernik have introduced a perturbation expansion in powers of ε for the longitudinal impedance of such a waveguide. We have reformulated the derivation of this expansion in a manner which elucidates the structure of the higher-order terms, and allows the determination of the dependence on ε of the resonant frequencies. For a square-wave (SW) wall distortion, there are divergent coefficients in the perturbation expansion. Hence, a special treatment is required in this case, and we present a calculation of the resonant behavior for small ε, using an approach which does not assume an expansion in powers of ε. The resulting expression for the resonant impedance involves functions singular at ε=O; however, to leading order in ε, the loss-factors and resonant frequencies are in agreement with the perturbation theory of ref. 1.