Wronskian flux ($\mathcal{J}=\mathrm{Im}(E^{\ast }{E}^{\prime })$)

For a complex solution of $E^{\prime \prime }+k^2(x)E=0$ with real $k^2$, the Wronskian fluxis constant: differentiation leaves $\mathrm{Im}(\overline{E}{E}^{\prime \prime })=-k^2\mathrm{Im}|E{|}^2=0$. The directed wave flux is therefore conserved even when the local wavenumber varies. In a one-dimensional ordinary plasma wave, the mean Poynting flux is $c^2\mathcal{J}/(8\pi \omega )$ in Gaussian units.

For constant positive asymptotic wavenumbers $k_{-},k_{+}$, write the left field as $e^{i{k}_{-}x}+r{e}^{-i{k}_{-}x}$ and the transmitted right field as $t{e}^{i{k}_{+}x}$. Their fluxes are $k_{-}(1-|r{|}^2)$ and $k_{+}|t{|}^2$. Conservation givesHere the conserved quantities are flux fractions. The complex amplitude coefficients themselves need not sum to1, and a transmission fraction need not equal $|t{|}^2$ when the channel normalizations differ.