Impedance of a rectangular beam tube with small corrugations

We consider the impedance of a structure with rectangular, periodic corrugations on two opposing sides of a rectangular beam tube. Using the method of field matching, we find the modes in such a structure. We then limit ourselves to the case of small corrugations, but where the depth of corrugation is not small compared to the period. For such a structure we generate analytical approximate solutions for the wave number k, group velocity v(g), and loss factor kappa for the lowest (the dominant) mode which, when compared with the results of the complete numerical solution, agreed well. We find if w similar to a, where w is the beam pipe width and a is the beam pipe half-height, then one mode dominates the impedance, with k similar to 1/rootwdelta (delta is the depth of corrugation), (1 - v(g)/c) similar to delta, and kappa similar to 1/(aw), which (when replacing w by a) is the same scaling as was found for small corrugations in a round beam pipe. Our results disagree in an important way with a recent paper of Mostacci et al. [A. Mostacci et al., Phys. Rev. ST Accel. Beams 5, 044401 (2002)], where, for the rectangular structure, the authors obtained a synchronous mode with the same frequency k, but with kappa similar to delta. Finally, we find that if w is large compared to a then many nearby modes contribute to the impedance, resulting in a wakefield that Landau damps.