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 κ for the lowest (the dominant) mode which, when compared with the results of the complete numerical solution, agreed well. We find if w∼a, where w is the beam pipe width and a is the beam pipe half-height, then one mode dominates the impedance, with k∼1/sqrt[wδ] (δ is the depth of corrugation), (1-v_g/c)∼δ, and κ∼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 κ∼δ. 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.