 Stationary wave amplitude in a stratified wavespeed

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.