High-frequency behavior of the longitudinal impedance for a cavity of general shape

Abstract

An ultrarelativistic beam bunch traveling along the axis of an azimuthally symmetric cavity of general shape connected to a beam pipe generates wake fields. An integral equation is derived for the longitudinal electric component of these wake fields at the beam-pipe radius. The kernel for this equation is approximated at high frequency for several different cavity shapes and the resulting integral equation is solved for the electric field at the beam-pipe radius. The longitudinal coupling impedance is obtained and shown to vary inversely as the square root of the frequency, in agreement with recent results obtained by others using different approximation techniques. The results are also shown to depend only on the axial extent of the cavity at the beam-pipe radius, but otherwise are independent of cavity shape for four different cavity geometries. In addition, the results are shown to be in agreement with the analytic properties of the coupling impedance as a function of complex frequency required by causality for an ultrarelativistic beam.