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Phys. Rev. ST Accel. Beams 6, 024401 (2003)

Impedance of a rectangular beam tube with small corrugations

K. L. Bane and G. Stupakov

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  17. There is another way, one depending on the beam energy, by which the wake effect in our structure can go to zero as δ→0. First, note that the wakefield typically (and also here) is defined as the response (longitudinal voltage per unit driving charge) experienced at distance s behind a driving point particle moving at infinite energy through a structure. The impedance is the Fourier transform of the wakefield. For a driving particle of finite (but high) energy γ the impedance will be almost the same, except that it will drop to zero for frequencies ω≳cγ/a. Now if at a finite, fixed energy, we let δ decrease, therefore causing the resonance frequency k of our structure to increase, we will eventually end up with k≳γ/a, at which point the impedance, and therefore the wake, will quickly drop to zero.
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  19. It probably is valid for the (steady-state) wake of any cylindrically symmetric, periodic structure, with a the closest approach of the structure to the beam axis.
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