Help understanding landau damping derivation

1. May 14, 2012

hereboy!!

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)] [1]

I understand it up till this point, the next part leaves me confused.

equation [1] 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.

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2. May 14, 2012

Bill_K

To simplify let Δ = ku0 - ω. Eq [1] is du/dt = e/m Eeikx0 (eiΔt). Integrating the parenthesis wrt t gives u - u0 = e/m Eeikx0 (eiΔt/iΔ). Then to evaluate the limit Δ → 0 we use l'Hopital's rule. Diffierentiate numerator and denominator of the () wrt Δ and get u - u0 = e/m Eeikx0 (it/i), which is what you have.

3. May 14, 2012

hereboy!!

Thanks for the speedy response, very much appreciated!

4. May 14, 2012

hereboy!!

one more thing i forgot to ask, what does u-u0 represent and how does this show that electrons with velocities u0 close to the phase velocity of the wave have pertubations that grow in time?

Best Wishes

G