High frequency dependence of the coupling impedance for a large number of obstacles

The author recently derived an integral equation for the axial electric field at the pipe radius in the presence of an azimuthally symmetric cavity of arbitrary shape in a beam pipe of circular cross section. In the present work, the author analyzes the case of N cavities and explores the high-frequency behavior for large N, in an effort to understand the transition to a periodic structure Not unexpectedly, the result depends critically on which of the limits (k→∞ or N→∞) is taken first