Beam loading effects for two-beam ring

An analytic treatment of multi-bunch potential well distortion for a two-beam storage ring is presented. The longitudinal wake effects are separated into: the mode loss, the synchrotron tune shift (both due to potential well distortion) and the coherent multi-bunch coupling. Here, only the first two effects are studied. Resulting simple analytic formulas describe the mode loss and the synchrotron tune shift experienced by a given bunch within the two-beam, as a function of the high order mode's parameters. One can get immediately a simple quantitative answer in term of the mode loss and the synchrotron tune shift experienced by each bunch from these formulas, so the authors can know how to modify the existing configuration of parasitic cavity resonance (via frequency tuning) so that the resulting potential well distortion effects are minimized. When the RF cavities are symmetrically distributed about the interaction points, the two beams will have same beam loading effects, so the authors can compensate the phase shift of the two beam using the same method as in one beam case