Transverse electron-proton two-stream instability in a bunched beam

This paper is an analytical investigation of the transverse electron-proton (e-p) two-stream instability in a proton bunch propagating through a stationary electron background. The equations of motion, including the effect of damping, are derived for the centroids of the proton beam and the electron cloud. An approach is developed to solve the coupled linear centroid equations in the time domain describing the e-p instability in proton bunches with nonuniform line densities. Examples are presented for proton line densities corresponding to uniform and parabolic profiles