A longitudinal plasma wave is set up propagating in the x direction (k = kx̂) in a plasma whose equilibrium state is characterized by the following so-called resonance distribution of velocities in the direction of the wave normal of the longitudinal plasma wave:

where A is a constant.
(a) Using this expression for f₀(v) in the dispersion relation for the longitudinal plasma wave, given in (4.9), obtain the result

(b) Evaluate the integral of part (a) by closing the integration contour in the upper half plane (note that there is a double pole at v_x = ω/k and a single pole at v_x = iA), to obtain the dispersion relation

(c) Analyze this dispersion relation (ω = ω_r + iω_i) to show that the longitudinal wave in this plasma is not unstable and determine the frequency of oscillation (ω_r) and the Landau damping constant (ω_i)· Compare this Landau damping constant with the corresponding value for a Maxwellian distribution of velocities, for the cases when kλ_D ≪ 1 and kλ_D ≥ 1.

Equation 4.9.

Transcribed Image Text:

fo(v) = no A πT 1 (v² + A²) 8(vy) 8 (vz)