• V. Balbekov
  Fermilab, Batavia, Illinois

Some results of theoretical investigations of transverse dipole instability of a rectangular bunch are reported in this paper. Such a form is characteristic of the bunch in a rectangular potential wall which is created by a barrier-shaped acceleration field. Similar regime is a major one for accumulating and cooling of antiproton beams in the Fermilab Recycler Ring. In this case, the known theory of transverse instability of a bunched beam is inapplicable directly both because of "unusual" form of phase trajectories and strong dependence of synchrotron frequency on energy. A series of equations, adequately describing the instability is derived in the paper. Exact analytical solution is obtained for space charge dominated impedance, and some approximate methods are proposed for arbitrary impedance. The theory is applied to the Fermilab Recycler Ring including a numerical simulation.