WKB amplitude in a slowly varying duct

For a separated duct mode with local width $h(X)$ and nonzero axial wavenumber $k(X)$, conservation of modal flux gives amplitude proportional to $[h(X)k(X){]}^{-1/2}$. The phase is ${ϵ}^{-1}{\int }^{X}k(\xi )d\xi$.