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For parallel propagation, use the undivided dispersion relation:The five requested cases are:

For the perpendicular cases the undivided dispersion relation factors as $(n^2-P)(S{n}^2-RL)=0$, up to an overall nonzero factor. This also identifies the extraordinary plasma wave plasma wave resonance at $S=0$.

For the ordinary plasma wave, choose $\mathbf{k}=k\hat{\mathbf{x}}$ and $\delta \mathbf{E}=E_z\hat{\mathbf{z}}$. The equations in a give $u_{xs}=u_{ys}=0$ andThe driven particle motion is entirely parallel to ${\mathbf{B}}_0$: the parallel Lorentz force has no magnetic contribution. Nevertheless, the ordinary plasma wave is electromagnetic, since $\delta \mathbf{E}\perp \mathbf{k}$ and the Faraday law givesAn electrostatic plasma wave instead has $\delta \mathbf{E}\parallel \mathbf{k}$ and zero perturbed magnetic field. Alignment with the equilibrium magnetic field does not make a wave electrostatic. At the exact plasma wave cutoff $k=0$, this propagating-wave distinction becomes degenerate.

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.

For real $k^2(x)$, the wave equation has the conserved Wronskian fluxFor this ordinary plasma wave, the time-averaged Poynting flux is $⟨S_x⟩=c^2\mathcal{J}/(8\pi \omega )$. With no incident wave from the right, evaluating the same Wronskian flux on both sides provesThe coefficients in f also satisfy this identity algebraically, before the opaque-barrier expansion.

A plasma wave resonance makes the ideal cold description singular and requires an appropriate causal regularization. A passive resonant layer can absorb wave energy into particle motion, or convert it into another branch. For the two measured outgoing channels one then haswith $A>0$ when resonant absorption or uncounted mode conversion occurs. A numerical value of $A$ requires the layer physics; a resonance by itself does not specify a universal loss fraction. Counting all converted and absorbed energy preserves conservation of energy.

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

For $E^{\prime \prime }+k^2(x)E=0$ with real $k^2$, differentiation gives ${\mathcal{J}}^{\prime }=\mathrm{Im}(E^{\ast }{E}^{\prime \prime })=0$. This conserved Wronskian measures directed wave flux. In a one-dimensional ordinary plasma wave, the mean Poynting flux is $c^2\mathcal{J}/(8\pi \omega )$ in Gaussian units. It proves $R+T=1$ for a lossless two-channel scattering problem.