Electron Acceleration in High-Frequency Longitudinal Waves, Doppler-Shifted Ponderomotive Forces, and Landau Damping

An electron in the field of a high-frequency longitudinal wave experiences both Doppler-shifted intensity-gradient and phase-gradient ponderomotive forces. It is shown that Landau damping can be considered as a phase-gradient force effect. By taking the wave amplitude space dependence into account, we demonstrate another collisionless effect due to the intensity-gradient force. Either damping or amplification of the wave can occur depending on whether the electron density increases or decreases.