Wave-action transport for a variable-speed wave ($(\omega A^2{)}_T+\nabla \cdot (c^2\mathbf{k}{A}^2)=0$)

The first-order imaginary balance of the wave equation in geometrical optics gives ${\partial }_T(\omega A^2)+\nabla \cdot (c^2\mathbf{k}{A}^2)=0$ for a real leading amplitude. The action density $\omega A^2$ travels with the group velocity $c^2\mathbf{k}/\omega$. For a complex amplitude replace $A^2$ by $|A{|}^2$.

For an individual monochromatic component in $c=c(Y)$, conserved $\omega ,k$ give $\ell =\sqrt{{\omega }^2/c^2-k^2}$. With amplitude independent of $X,T$, wave-action transport for a variable-speed wave conserves $c^2\ell A^2$ on a positive-$\ell$ branch, hence $A=\mu /(c\sqrt{\ell })$. The divergence at $\ell =0$ signals failure of the outer WKB approximation.