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Write the ordinary plasma wave asThe eikonal function $S$ is dimensionless; it varies by order ${\epsilon }^{-1}$ across a background scale $L_{\mathrm{bg}}$, where $\epsilon \sim \lambda /L_{\mathrm{bg}}\ll 1$. The amplitudes and equilibrium plasma parameters vary on $L_{\mathrm{bg}}$, so $|\nabla {\mathbf{a}}_j|\ll k|{\mathbf{a}}_j|$ and $|\nabla k|\ll k^2$. The equilibrium is stationary, so $\omega$ is constant along propagation. Assume infinitesimal fields, negligible collisions, $k{v}_{\mathrm{th},s}\ll \omega$, negligible ion response, and a fixed ordinary plasma wave polarization with ${\mathbf{B}}_0\parallel \hat{\mathbf{z}}$ and spatial variation along $x$. The WKB approximation is used away from wave turning points and plasma wave resonances.

Replacing derivatives on the fast factor by $i\mathbf{k}$ and $-i\omega$ in the Maxwell equations reproduces the local magnetized cold-plasma dielectric tensor equation of b. In this one-dimensional ordinary plasma wave geometry, the Maxwell equations and the local cold parallel current density in fact give the scalar equation directly:For $\delta E_z=a(x)e^{iS(x)}$, substitution givesThe leading eikonal equation is $(S^{\prime }{)}^2=k^2(x)$, and the next order gives $2k{a}^{\prime }+k^{\prime }a=0$, hence the flux-normalized WKB amplitude $a\propto k^{-1/2}$. In particular, require $|k^{\prime }|/|k{|}^2\ll 1$ and $|k^{\prime \prime }|/|k{|}^3\ll 1$; these fail at the wave turning points.

A wave turning point satisfies $k^2(x)=0$, equivalently ${\omega }_{pe}^2(x)={\omega }^2$. ThereforeFor $0<\omega <{\omega }_{pe0}$ 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

At $a=-x_T$, defineThe first-order Taylor expansion is $k^2(x)=-\alpha (x-a)+O((x-a{)}^2)$, so the local wave equation becomesThis is the Airy ordinary differential equation. Its $z<0$ side is oscillatory and its $z>0$ side is evanescent. A matching overlap requires $\ell \ll |x-a|\ll x_T$, hence well-separated wave turning points with $\ell \ll x_T$. At $b=x_T$, the coordinate $z_R=(b-x)/\ell$ gives the same Airy ordinary differential equation, with its evanescent side again at positive argument.

Let $J(x)={\int }_x^b\kappa (s)ds=\eta -I(x)$. The WKB approximation in the barrier can be rewritten asThe right Airy turning-point connection formula givesso $A_{\mathrm{t}}=D_R+i{G}_R/2$ and $A_{\mathrm{back}}=D_R-i{G}_R/2$. Impose no incident wave from the right, $A_{\mathrm{back}}=0$. ThenConsequently, within the leading Airy turning-point connection formula calculation,For an opaque barrier, $\eta \gg 1$,The transmission coefficient is a ratio of Poynting fluxes; the $k^{-1/2}$ normalization already includes their local wavenumber factors. The displayed denominator is useful for consistent matching and Wronskian flux conservation, but does not claim accuracy for subleading terms beyond the WKB approximation, or for merging wave turning points.

The barrier wavenumber is $\kappa ={\omega }_{pe0}\sqrt{x_T^2-x^2}/(cL)$. The substitution $x=x_T\sin u$ givesThus the leading transmission coefficient isAt 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 $T\to 1$ 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.

The shaded area in the lower panel is $\eta$ after restoring the factor ${\omega }_{pe0}L/c$.

A zero of the local squared wavenumber in $E^{\prime \prime }+k^2(x)E=0$. At a simple zero, $(k^2{)}^{\prime }(x_T)\ne 0$, the WKB approximation fails in a layer with scale $|(k^2{)}^{\prime }(x_T){|}^{-1/3}$ and the Airy ordinary differential equation supplies the local replacement. Two nearby wave turning points require a uniform treatment rather than independent Airy turning-point connection formulas.