= Solution
The barrier <wavenumber> is $\kappa=\omega_{pe0}\sqrt{x_T^2-x^2}/(cL)$. The substitution $x=x_T\sin u$ gives
$$
\begin{aligned}
\eta&=\frac{\omega_{pe0}}{cL}\int_{-x_T}^{x_T}\sqrt{x_T^2-x^2}\,dx
=\frac{\omega_{pe0}x_T^2}{cL}\int_{-\pi/2}^{\pi/2}\cos^2u\,du\\
&=\boxed{\frac{\pi\omega_{pe0}L}{2c}\left(1-\frac{\omega^2}{\omega_{pe0}^2}\right).}
\end{aligned}
$$
Thus the leading <transmission coefficient> is
$$
\boxed{T\simeq\exp\!\left[-\frac{\pi\omega_{pe0}L}{c}\left(1-\frac{\omega^2}{\omega_{pe0}^2}\right)\right].}
$$
At fixed $\omega/\omega_{pe0}$, increasing the column width or peak <plasma frequency> suppresses transmission exponentially. Raising $\omega$ towards $\omega_{pe0}$ narrows the evanescent region and increases transmission. However, when $\eta$ becomes order one or smaller the two <wave turning points> interact; the opaque-barrier <WKB approximation> cannot be extrapolated to claim $T\to1$ at $\omega=\omega_{pe0}$. Smooth matching to an exterior is also implicit in the leading result; at $|x|=L$ the stated profile joins continuously to <vacuum>.
\Image[/past-exam-of-the-collisionless-plasma-physics-course-of-the-university-of-oxford/original2025-tunnelling.png]
{title=Local squared <wavenumber> and evanescent barrier for an <ordinary plasma wave>}
The shaded area in the lower panel is $\eta$ after restoring the factor $\omega_{pe0}L/c$.
Back to article page
