Cumulative beam breakup with smoothly varying parameters

The difference equations for cumulative beam breakup are converted to partial differential equations by identifying slowly varying parameters and neglecting rapidly varying terms. The equation thus obtained leads to a solution for constant parameters that is identical to the transient solution obtained earlier. Solutions are then obtained for slowly varying parameters as a perturbation expansion in powers of the rate of change of these parameters. Analytic results are presented for slowly varying energy, slowly varying charge per bunch, slowly varying deflecting-mode frequency (such as might occur in a relativistic proton linac), and smooth external focusing. Smooth variation of the charge per bunch will suppress the beam-breakup transient if there is a systematic initial displacement. Variation of deflecting-mode frequency will also depress the beam-breakup transient by increasing the value of 1/Q to that corresponding to the relative width of the frequency variation. The presence of focusing also depresses the beam breakup, and a threshold parameter for this reduction is presented. All analytic results are shown to be in close agreement with those obtained using simulations. (author) 9 refs., 4 figs