[en] Electrostatic bounce modes are standing waves that occur in a mirror plasma when the relative spread in electron bounce frequencies is small. The modes can be destabilized by an ion distribution with a peaked perpendicular energy, and experimental data suggest that this mechanism was the principal cause of instability in certain low-density mirror experiments. After a review of theoretical work on electrostatic waves in mirror plasmas, a general matrix eigenvalue equation for the wave potential is derived which accounts accurately for electron histories and which includes the ion response. A computer program for calculating the plasma eigenmodes and the associated threshold densities for instability and maximum growth rates is then described. The threshold densities for unstable bounce modes expected in the Baseball I and Baseball II devices are compared with experimental values. The good agreement between theoretical and experimental thresholds in Baseball II makes bounce modes the most likely cause of instabilities in that device. In Baseball I, the most unstable modes expected from the theory have threshold densities consistently below observed values