Summary and discussions

In the above, we calculate the average power absorbed by a group of resonant particles moving in a longitude wave. The result [Eq. (39)] indicates that (1) if $\omega / k > 0$ and $\left[ \frac{d f (v_0)}{d v_0} \right]_{v_0 = \omega / k} < 0$, then the power is positive, which means the particles get energy from the wave, which further means the wave are damped. (2) if $\omega / k < 0$ and $\left[ \frac{d f (v_0)}{d v_0} \right]_{v_0 = \omega / k} > 0$, then the power is also positive, which also means the wave are damped. The two cases [(1) and (2)] can be summarized in a simple sentence: If there are more resonant particles moving slower than the wave phase velocity than those moving faster, then the wave is damped (where the resonant particles refer to the particles with $v \approx \omega / k$).

The result given above is obtained in the test particle approximation, which means the wave is given and is not necessarily a self-consistent field.