Coupling impedances of small discontinuities: A general approach

A general theory of the beam interaction with small discontinuities of the vacuum chamber of an accelerator is developed taking into account the reaction of radiated waves back on the discontinuity. The reactive impedance calculated earlier is reproduced as the first order and the resistive one as the second order of a perturbation theory based on this general approach. The theory also gives, in an easy and natural way, the analytical results for the frequencies and coupling impedances of the trapped modes due to small discontinuities on the vacuum chamber of a general cross section. Formulas for two important particular cases---a circular and a rectangular chamber---are presented.