Derivation of azimuthal harmonics series for stability analysis of the coherent oscillations of bunched beams

Stability analysis of the coherent motion of multibunch beams (including counter-rotating beams) should involve expressions analogous to the effective impedance—the series over harmonics of the revolution frequency of the impedance of the RF structure at the side frequencies to these harmonics—with certain factors depending on the harmonic number, such as the bunch line-density spectrum, the phase factor and the factor describing the order of multipole synchrotron oscillations. In this paper, we present a method for analytical summation of these series for resonant impedance, which does not seem to have been carried out previously for the general case including all the factors mentioned. Comparison of the expressions obtained with formulae used in previous papers shows the limits of validity of simpler approaches. The expressions obtained are used in the computer codes MBIM1 and MBIM2 presented at this conference, which calculate coherent oscillation stability for arbitrary multibunch beams.