 Solution

For a stationary component, the stationary internal-wave WKB solution has local vertical wavenumber and vertical-velocity amplitudewith the upward branch $m>0$. The WKB approximation requires $|m^{\prime }|\ll m^2$ and sufficient smoothness, with $|m^{\prime \prime }|\ll |m{|}^3$ a useful additional criterion. The background changes little over a vertical wavelength; this excludes a turning level of a stationary internal wave.

For increasing $N=N_0(1+z/H)$, the vertical wavenumber increases. The crest slope $-k/m$ tends toward zero, and successive crests get closer vertically. The laboratory energy ray obeys $dx/dz=k/m$, so it becomes more nearly vertical as it rises. There is no upper turning level in this ideal increasing-stratification profile.

For decreasing $N=N_0(1-z/H)$, the vertical wavenumber decreases to zero atThis assumes propagation at the bottom, $k{U}_0<N_0$. The crests become more nearly vertical and the incident energy ray becomes horizontal at $z_t$. Above it the wave is evanescent, and the incident wave is reflected downward. The divergent WKB amplitude at $m=0$ is an approximation failure, resolved by an Airy function transition, rather than an infinite physical wave amplitude. The decreasing formula for $N$ is only a positive-frequency background below $z=H$; the relevant turning height lies below that level.