Longitudinal collective modes of a bunched beam with a repulsive inductive impedance (the space charge below transition or the chamber inductance above it) are analytically described by means of reduction of the linearized Vlasov equation to a parameterless integral equation.For any multipolarity, the discrete part of the spectrum is found to consist of infinite number of modes with real tunes, which limit point is the incoherent zero-amplitude frequency.In Foot Wear other words, notwithstanding the rf bucket nonlinearity and potential well distortion, the Landau damping is lost.Hence, even a tiny coupled-bunch interaction Oral Care makes the beam unstable; such growth rates for all the modes are analytically obtained for arbitrary multipolarity.In practice, however, the finite threshold of this loss of Landau damping is set either by the high-frequency impedance roll-off or intrabeam scattering.
Above the threshold, growth of the leading collective mode should result in persistent nonlinear oscillations.