Method for finding bunched beam instability thresholds

At high intensity, a short range wake field can distort the beam's potential well and thereby change the stationary distribution. It is now known that if this is not taken into account, instability thresholds will be incorrectly predicted. A numerical method exists for solving the linearized Vlasov equation for the self-consistent case, including the distortion to the stationary distribution, and finding such thresholds. We have found physical explanations for the eigenmodes and instability thresholds predicted in this method. As a result, a much simpler stability criterion has been found. The criterion is simple in that it depends only on the stationary distribution and does not require solution of the linearized Vlasov equation. (author)