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Head-tail modes in bunches with space charge are studied using particle tracking simulations. The eigenfrequencies and eigenfunctions of transverse coherent oscillations in a Gaussian bunch are determined and compared with theories. A model for an airbag distribution in a barrier potential gives good predictions for the head-tail spectrum and for eigenfunctions in bunches with space charge. Using numerical simulations, space-charge induced Landau damping in a bunch is demonstrated. The damping rates are quantified for different modes and space-charge tune shifts. Finally, the head-tail instability with space charge is studied for the resistive-wall impedance below the mode coupling threshold. Results demonstrate that space-charge induced damping can suppress the instability for moderately strong space charge; instability growth rates saturate at strong space charge, in agreement with theoretical predictions

Head-tail instability and Landau damping in bunches with space charge