 Solution

ID: past-exam-of-the-collisionless-plasma-physics-course-of-the-university-of-oxford/2025/2/b/solution

k^{2} (x) = 0

, equivalently

ω_{p e}^{2} (x) = ω^{2}

. Therefore

x = \pm x_{T}, x_{T} = L 1 - \frac{ω ^{2}}{ω _{p e 0}^{2}} .

For

0 < ω < ω_{p e 0}

the two wave turning points lie inside the column and are simple. Defining

a = - x_{T}

and

b = x_{T}

, the local dispersion relation becomes

k^{2} (x) = \frac{ω _{p e 0}^{2}}{c ^{2} L ^{2}} (x^{2} - x_{T}^{2}) .

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