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Cumulative beam break-up in a periodic linac.

The problem of cumulative beam break-up for a coasting beam in a linac with periodically placed acceleration cavities and transverse focusing units is treated using a general technique. The equations of motion for a series of equally spaced pointlike bunches are reduced to a single Hill's-type differential equation, from which a transfer matrix for a single period is derived. The technique is applied to the special case of a single resonance impedance, yielding a result which is in agreement with that derived elsewhere. The case of two nearly degenerate resonances is investigated using saddle-point integration, in the limit of strong focusing.