I have been given a derivation describing the physics of landau damping, but i don't quite understand it. It starts with the equation for a charged particle in a 1d electric field varying as Eexp[i(kx-wt)] being determined by d2x/dx2=e/m Eexp[i(wt-kx)]. Since we are dealing with a linearized theory in which the perturbation due to the wave is small, it follows that if the particle starts with velocity u0 at position x0 then we may substitute x0+u0 t for x in the electric field term. This is actually the position of the particle on its unperturbed trajectory, starting at x = x0 at t = 0. Thus, we obtain du/dt=e/m Eexp[i (kx0+ku0 t−ωt)]  I understand it up till this point, the next part leaves me confused. equation  yields: u − u0 =e/m E [[exp i(kx0+ku0 t−ωt) − exp ikx0)]/i(ku0 − ω)] As k u0 − ω → 0, the above expression reduces to u − u0 =e/m Et exp ik x0 showing that particles with u0 close to ω/k, that is with velocity components along the x-axis close to the phase velocity of the wave, have velocity perturbations which grow in time. These so-called resonant particles gain energy from, or lose energy to, the wave, and are responsible for the damping. I have attached the derivation that is in my textbook as a seperate file, If anyone could point me in the right direction to understanding this it would be greatly appreciated.