= Wronskian flux
{c}
{title2=$\mathcal J=\operatorname{Im}(E^*E')$}
For a complex solution of $E''+k^2(x)E=0$ with real $k^2$, the <Wronskian flux>
$$
\mathcal J=\operatorname{Im}(\overline E E')
$$
is constant: differentiation leaves $\operatorname{Im}(\overline E E'')=-k^2\operatorname{Im}|E|^2=0$. 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 $e^{ik_-x}+r e^{-ik_-x}$ and the transmitted right field as $t e^{ik_+x}$. Their fluxes are $k_-(1-|r|^2)$ and $k_+|t|^2$. Conservation gives
$$
\mathcal R+\mathcal T=1,\qquad\mathcal R=|r|^2,\quad\mathcal T=\frac{k_+}{k_-}|t|^2.
$$
Here 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.
Back to article page
