Longitudinal effects of near-uniform beam-wall wake fields

Several theories have been developed in the past to explain longitudinal instabilities of individual bunches in particle accelerators and storage rings. But they are less than satisfactory because in one way or another they rely on some mathematical approximation of doubtful physical meaning. Here, the basic physical model is simplified in order to help to understand a little better the physics involved. Initially, the motion of two particles executing phase oscillations in the same bunch under the influence of the wake field generated by the other is investigated. This motion is stable and bounded even for the case of a slowly varying wake field. But the distortions of the trajectories can, nevertheless, be significant and can be interpreted as bunch lengthening and widening. These models generalized to an N particle model. If the wake field is constant, the equations are simple enough that it is possible to solve exactly for the intensity-limited, self-consistent, longitudinal particle distribution. It is not necessary to solve, or even introduce, the Vlasov equation. These models exhibit effects like dipole and quadrupole oscillations and bunch lengthening, in qualitative agreement with observed behavior, but no attempt has been made at quantitative comparison with observation