Longitudinal coupling impedance of a small obstacle

The longitudinal impedance for a narrow pillbox with beam pipes was investigated analytically and numerically by the authors recently (1987) in order to understand the structure of the impedance as a function of frequency in the region above the cutoff of the beam pipe. The field matching procedure described by H. Henke (1985) was used in the limit of a narrow pillbox, and the impedance was expressed as a sum which required a cutoff for convergence. The authors start with the integral equation for the field at the pipe radius and solve this equation rigorously in the case of a small obstacle of general shape. In this way, they obtain a more accurate representation of the important broad resonance which dominates the behavior in the frequency range above the cutoff of the beam pipe