Bunch motion in the presence of the self-induced voltage due to a reactive impedance; 1, RF off

Analytic self-consistent solutions have been found for the nonlinear Vlasov equation describing different types of behaviour with time of an intense bunch under the influence of voltage induced due to a reactive part of broad band impedance. The problem is solved for the particular type of the initial distribution function in longitudinal phase space which is elliptic and corresponds to parabolic line density. The first part of the paper is devoted to the consideration of the effects in the machine with RF off. In this case induced voltage is changing with time and, as in the case with RF on, can have a significant effect on bunch motion. Numerical estimations for the SPS show that this effect can be important for manipulations with beam at 26GeV. Measurements of the change in the rate of debunching with intensity can also be used to estimate the value of the impedance. The same method is applied in the second part of the paper to analyse time dependent effects of potential well distortion when RF is on.