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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 ${\omega }_{pe0}L/c$.

Balance $ϵy^{\prime \prime }$ against $xy$ near the wave turning point. With $x=\gamma X$, their equality requires $ϵ/{\gamma }^2=\gamma$, hence $\gamma ={ϵ}^{1/3}$. The rescaled equation isIts leading equation is the Airy ordinary differential equation $Y^{\prime \prime }=XY$. ThusLet $a_0=\mathrm{Ai}(0)=3^{-2/3}/\Gamma (2/3)$; then $\mathrm{Bi}(0)=\sqrt{3}{a}_0$. The left boundary condition becomes $a_0(\alpha +\sqrt{3}\beta )=1$, orThis determines one linear relation before the far-boundary matching condition is used.

Choose $\delta =\sqrt{ϵ}$ in the WKB approximation and substitute $y=\exp (S_0/\delta +S_1+\cdots )$. The leading equation isso take $S_0=\pm 2x^{3/2}/3$. At the next order,givingup to an absorbed constant. Label the recessive and dominant branches respectively byThe formal outer solution is $c_1{y}_1+c_2{y}_2$. The point zero is a simple wave turning point: $S_0^{\prime }$ vanishes and the WKB amplitude diverges.

The boundary condition at infinity selects $c_2=0$, with $c_1$ to be fixed by matching. This statement concerns the actual recessive branch. One cannot literally send $x\to \infty$ in the truncated small-$ϵ$ formulas: their ordering fails when $ϵx$ is no longer small, and their common Gaussian factor would then spuriously suggest that both branches decay.

To justify the selection, remove the first derivative exactly with $y=e^{-x^2/4+x/2}v$. ThenThe coefficient is positive for $x\ge 0$, so it has distinct dominant and recessive branches. For very large $x$, the dominant branch restores $y$ growth of exponential rate $1/ϵ$, while the recessive branch decays. Matching to this recessive branch removes $c_2$.

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.

The WKB approximation separates rapid phase variation from a slowly varying amplitude in a linear ordinary differential equation. Foron an interval with smooth $q>0$, its leading oscillatory form isSubstitution of $y=A{e}^{iS/\epsilon }$ gives $S^{\prime 2}=q$ at leading order and $2S^{\prime }{A}^{\prime }+S^{\prime \prime }A=0$ at the next order, hence $A\propto q^{-1/4}$. This is the mathematical origin of the flux-normalized WKB amplitude.

Where $q<0$, the leading branches instead grow and decay exponentially, with $|q{|}^{-1/4}$ amplitude and exponents $\pm {\epsilon }^{-1}\int \sqrt{|q|}dx$. A wave turning point at $q=0$ makes this formula singular and requires an Airy turning-point connection formula for a simple zero. Slow variation and an error estimate are essential; the displayed expression is not an exact solution for arbitrary $q$. The real exponential version has a rigorous error-control theorem.^[ref]