Past exam of the collisionless plasma physics course of the University of Oxford / 2025 / 2 / c / Solution

Use positive real local wavenumberswithin $|x|\le L$. If the exterior is vacuum, continue $k_1,k_3$ as $\omega /c$ outside the column. Define the positive phase integrals and barrier integralThe three WKB approximations, with the common $e^{-i\omega t}$ suppressed, areThe flux-normalized WKB amplitudes make squared travelling-wave coefficients proportional to directed Poynting flux. Since ${\Phi }^{\prime }=-k_1$ and ${\Psi }^{\prime }=k_3$, $A_{\mathrm{in}}$ and $A_{\mathrm{t}}$ travel to the right, while $A_{\mathrm{r}}$ and $A_{\mathrm{back}}$ travel to the left. The constants $\pi /4$ are a convenient Airy turning-point connection formula convention.