Wave damping

If the longitudinal wave is an electron plasma wave, then the wave energy consists of two components, the energy of the electric field and the averaged kinetic energy of the particle oscillations,

$$W = W_E + W_p,$$ (40)

where $W_E$ is the energy density of the electric field averaged in one wavelength, which is given by

$$W_E = \frac{k}{2 \pi} \int_0^{2 \pi / k} \frac{\varepsilon_0}{2} [E \cos (k x)]^2 d x = \frac{1}{4} \varepsilon_0 E^2,$$ (41)

$W_p$ is the averaged kinetic energy of the particle oscillations, which, for electron plasma wave, is equal to the electric field energy $W_E$[3]. Using these results, equation (40) is written

$$W = \frac{1}{2} \varepsilon_0 E^2 .$$ (42)

The energy conservation requires that the kinetic energy gained by the resonant particles must come from the wave energy, i.e.,

$$\frac{d W}{d t} = \frac{\pi \omega}{\vert k\vert k} \frac{(q E)^2}{2 m} \left[ \frac{d f (v_0)}{d v_0} \right]_{v_0 = \omega / k} .$$ (43)

Using Eq. (42), equation (43) can be written

$$\frac{d E}{d t} = \frac{\pi \omega \omega_p^2}{2\vert k\vert k} \frac{1}{n_0} \left[ \frac{d f (v_0)}{d v_0} \right]_{v_0 = \omega / k} E,$$ (44)

where $\omega_p = \sqrt{n_0 q^2 / m \varepsilon_0}$ is the electron plasma frequency. Define

$$\gamma = \frac{\pi \omega \omega_p^2}{2\vert k\vert k} \frac{1}{n_0} \left[ \frac{d f (v_0)}{d v_0} \right]_{v_0 = \omega / k}$$ (45)

then Eq. (44) is written

$$\frac{d E}{d t} = \gamma E,$$

which can be integrated to give

$$E (t) = E (0) e^{\gamma t} .$$ (46)

The damping rate of the amplitude of the electric field given by Eq. (45) agrees the Landau damping in the weak growth rate approximation (equation (8-19) in Stix's book[4]).