Beam dynamics in a double RF system

The addition of a higher harmonic RF system to the main system allows a control of the synchrotron frequency, the spread in synchrotron frequency and the bunch length. Adjustment of the higher harmonic system so as to reduce the slope of the RF wave to zero at the bunch centre leads to a longer bunch and a greatly increased spread in synchrotron frequency. This increases the Landau damping against longitudinal coupled bunch instabilities. The motion of single particles in this highly non-linear potential is calculated numerically as well as analytically (by making some approximations). The dependence of the synchrotron frequency on amplitude and the forms of the synchrotron oscillations and the RF bucket are calculated. Finally the bunch shape and the distribution of particles in Q/sub s/ are calculated for electron bunches. (9 refs).