Longitudinal impedance of a periodic structure at high frequency

An ultrarelativistic beam bunch traveling along the axis of an azimuthally symmetric cavity of general shape connected to a beam pipe generates wake fields. We extend an earlier derivation of the integral equation for the axial electric field at the beam-pipe radius to the case of a periodic structure. This equation is then solved in the high-frequency limit, and we show that the reciprocal of the impedance per cell has a particularly simple form, closely related to the admittance for a single cavity. We also show that the imaginary part of the impedance per cell varies as ω^-1 and the real part of the impedance per cell varies as ω^-3/2. Finally, the results are shown to satisfy the requirements of causality for an ultrarelativistic beam.