= Solution
Write the <ordinary plasma wave> as
$$
\delta\mathbf E(\mathbf r,t)=\operatorname{Re}\{[\mathbf a_0+\varepsilon\mathbf a_1+\cdots]e^{iS(\mathbf r,t)}\},\qquad
\mathbf k=\nabla S,\quad\omega=-\partial_tS.
$$
The <eikonal function> $S$ is dimensionless; it varies by order $\varepsilon^{-1}$ across a background scale $L_{\rm bg}$, where $\varepsilon\sim\lambda/L_{\rm bg}\ll1$. The amplitudes and equilibrium <plasma> parameters vary on $L_{\rm bg}$, 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>, $kv_{\mathrm{th},s}\ll\omega$, 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 <1/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:
$$
\boxed{\frac{d^2\delta E_z}{dx^2}+k^2(x)\delta E_z=0,\qquad
k^2(x)=\frac{\omega^2-\omega_{pe}^2(x)}{c^2}.}
$$
For $\delta E_z=a(x)e^{iS(x)}$, substitution gives
$$
a''+2ik a'+ik'a+[k^2(x)-(S')^2]a=0.
$$
The leading <eikonal equation> is $(S')^2=k^2(x)$, and the next order gives $2ka'+k'a=0$, hence the <flux-normalized WKB amplitude> $a\propto k^{-1/2}$. In particular, require $|k'|/|k|^2\ll1$ and $|k''|/|k|^3\ll1$; these fail at the <wave turning points>.
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