The single-mode CSR instability for a bunched beam

The coherent synchrotron radiation (CSR) instability at the shielding threshold may be driven by a single synchronous mode excited by the beam in the beam pipe. The instability in this case has been analyzed [1] in the coasting beam approximation neglecting synchrotron motion. The later becomes important at large time intervals in storage rings where it substantially affects the beam dynamics. The single-mode CSR instability of a bunched beam with the synchrotron motion taken into account is described in this paper both in linear and nonlinear regimes. Analysis is relevant to other instabilities where the interaction is dominated by a single mode.