I have an equation (from the dispersion relation) for $w$ (frequency) in terms of $a$ (plasma frequency), $k$ (wave vector) and $v$ (velocity of the particles):
$$w^4-(a^2+2k^2v^2)w^2+k^2v^2(k^2v^2-a^2) = 0.$$
- How to find the wave's maximum growth rate?
- Why it is imaginary?
According to the given answer, the growth rate of this wave is imaginary and given as $$\Im\left(\frac wa\right)= \frac1{2\sqrt{2}}.$$