Charge Passing off-Axis through a Cylindrical Resonator with Beam Pipes

A charge is considered which passes off-axis through a cylindrical resonator with two semi-infinite beam pipes. In the pipe region the fields are represented by the source fields plus a continuous spectrum of waveguide modes and in the resonator region by an infinite, discrete set of resonator modes. Matching the fields at the common interface yields a set of linear equations for the expansion coefficients. The involved integrals are solved with the residuum calculus, where the poles are located so that only waves travelling away from the resonator region are taken into account. By truncating the system, numerical results were derived for the longitudinal and transverse impedance which, contrary to periodic systems, also have real parts due to radiation into the pipes.