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Help understanding landau damping derivation

  1. May 14, 2012 #1
    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.

    Attached Files:

  2. jcsd
  3. May 14, 2012 #2


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    Science Advisor

    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.
  4. May 14, 2012 #3
    Thanks for the speedy response, very much appreciated!
  5. May 14, 2012 #4
    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

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