Evolving bunch and retardation in the impedance formalism
The usual expression for the longitudinal wake field in terms of the impedance is exact only for the model in which the source of the field is a rigid bunch. To account for a deforming bunch one has to invoke the \it complete impedance, a function of both wave number and frequency. A computation of the corresponding wake field would be expensive, since it would involve integrals over frequency and time in addition to the usual sum over wave number. We treat the problem of approximating this field in an example of current interest, the case of coherent synchrotron radiation (CSR) in the presence of shielding by the vacuum chamber.