Amplification of longitudinal plasma oscillations upon a decrease in plasma concentration

Amplification of plasma oscillations in a homogeneous plasma by reducing its concentration is considered. The frequency of plasma oscillations decreases upon a decrease in the plasma concentration, and the resonant velocity of the plasma electrons, which is equal to the wave phase velocity, also decreases. Due to this, the number of resonant electrons exponentially increases in the equilibrium plasma. Since the energy of plasma oscillations is mainly determined by the contribution of the resonant electrons, this energy also increases exponentially.