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Coherent transverse oscillations of a bunch in a ring accelerator areconsidered. Three different models of the bunch are used: a hollow bunch in asquare potential well, a square bunch in a parabolic potential well, and aparabolic bunch in a parabolic well. The bunch space-charge field, wakefield,and chromaticity are taken into account in all the cases. The bunch eigenmodesand various instabilities are studied. The results are compared and analyzed toestablish the range of applicability of the models. Role of the Landau dampingdue to dependence of the SC field on the longitudinal coordinate is discussed.

Coherent transverse instability of bunched beams in circular accelerators