 Solution

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

Use positive real local wavenumbers

k_{1} (x) = \frac{ω _{p e 0}}{c L} x^{2} - x_{T}^{2} (x < a), k_{2} (x) = κ (x) = \frac{ω _{p e 0}}{c L} x_{T}^{2} - x^{2} (a < x < b), k_{3} (x) = \frac{ω _{p e 0}}{c L} x^{2} - x_{T}^{2} (x > b),

within

∣ x ∣ \leq L

. If the exterior is vacuum, continue

k_{1}, k_{3}

ω / c

outside the column. Define the positive phase integrals and barrier integral

Φ (x) = \int_{x}^{a} k_{1} (s) d s, I (x) = \int_{a}^{x} κ (s) d s, Ψ (x) = \int_{b}^{x} k_{3} (s) d s .

The three WKB approximations, with the common

e^{- iω t}

suppressed, are

E_{I} E_{II} E_{III} = \frac{A _{in} e ^{- i (Φ - π /4)} + A _{r} e ^{i (Φ - π /4)}}{k _{1}}, = \frac{D e ^{- I} + G e ^{I}}{κ}, = \frac{A _{t} e ^{i (Ψ - π /4)} + A _{back} e ^{- i (Ψ - π /4)}}{k _{3}} .

Φ^{'} = - k_{1}

and

Ψ^{'} = k_{3}

A_{in}

and

A_{t}

travel to the right, while

A_{r}

and

A_{back}

travel to the left. The constants

π /4

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