Longitudinal Instabilities of Long Gaussian Bunches

We present an overview of the longitudinal instabilities of Gaussian bunches subject to a harmonic RF potential. Our emphasis is on the behavior of long bunches having lengths greater than the wavelength of the perturbing electromagnetic fields. We exhibit the crossover between the dominance of the synchrotron modes and the coasting-beam-like distortions of the bunch distribution, which occurs as the real or imaginary part of the coherent oscillation frequency becomes large compared to the synchrotron oscillation frequency. For a narrow band impedance the growth rate of the coasting-beam-like modes is determined by the average beam current, and for a broad band impedance the growth rate is determined by the peak current. We discuss the transition between these two regimes by considering the growth rate as a function of the bandwidth of the impedance.