 Solution

ID: past-exam-of-the-kinetic-theory-course-of-the-university-of-oxford/2025/2/c/solution

For

K = k^{2} λ_{De}^{2} ≫ 1

, the logarithm in b must be large and positive. In each allowed interval this happens near a denominator zero, approached from the undamped side:

u_{-} = - v_{0} - O (e^{- K}), u_{a} = v_{1} + O (e^{- K}), u_{+} = V + O (e^{- K}) .

The constants hidden by

O (e^{- K})

have velocity units. Consequently,

ω_{-} \sim - k v_{0}, ω_{a} \sim k v_{1}, ω_{+} \sim k (v_{0} + v_{2} - v_{1}) .

The two outer modes have phase velocities just outside the occupied range and therefore have no resonant particles. The middle electron acoustic wave lies just inside the lower edge of the velocity plateau, where

f_{0}^{'} = 0

. All three escape Landau damping in this collisionless model. The stated neglect of ion response remains necessary for the middle mode as well.

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