Beam Breakup with Smooth Rapid Current Build-Up

Cumulative beam breakup in a high current linac can be represented by a set of difference equations for the Mth beam bunch in the Nth cavity. We here investigate the modification of the solution of these equations when the displaced beam current grows smoothly to its final value. Simulations show a dramatic reduction in the transient, even when the current growth takes place over only a few bunches. In our analysis, adiabatic results for large M and N are given for an exponential build-up of current corresponding to a "time constant" of T bunches. The solutions confirm the behavior observed in the simulations in all details, including the location of the transient peak, as well as the dramatic dependence of the magnitude of the transient on T.