Impedance of a rectangular beam tube with small corrugations

Abstract

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 $\ensuremath{\kappa}$ for the lowest (the dominant) mode which, when compared with the results of the complete numerical solution, agreed well. We find if $w\ensuremath{\sim}a$, where $w$ is the beam pipe width and $a$ is the beam pipe half-height, then one mode dominates the impedance, with $k\ensuremath{\sim}1/\sqrt{w\ensuremath{\delta}}$ ($\ensuremath{\delta}$ is the depth of corrugation), $(1\ensuremath{-}{v}_{g}/c)\ensuremath{\sim}\ensuremath{\delta}$, and $\ensuremath{\kappa}\ensuremath{\sim}1/(aw)$, 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 $\ensuremath{\kappa}\ensuremath{\sim}\ensuremath{\delta}$. 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.