Wake potentials and impedances of charged beams in gradually tapering structures

We develop an analytical method for calculating the geometric wakefield and impedances of an ultrarelativistic beam propagating on- and off-axis through an axially symmetric geometry with slowly varying circular cross-section, such as a transition. Unlike previous analytical methods, our approach permits detailed perturbative investigation of geometric wakefields, and detailed perturbative investigation of impedance as a function of frequency. We compare the accuracy of the results of our approach with numerical simulations performed using the code ECHO and determine parameters in which there is good agreement with our asymptotic analysis.