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Start from $\rho ({\partial }_{t}u+u\cdot \nabla u)=-\nabla p-\rho g{e}_z$, ${\partial }_t\rho +u\cdot \nabla \rho =0$, and $\nabla \cdot u=0$. The stationary background obeys $p_0^{\prime }=-g{\rho }_0$. Linearize in small perturbations and use the Boussinesq approximation: replace the density by a reference value ${\rho }_{\ast }$ in inertia but retain the density perturbation in the buoyancy force. With $\pi =p^{\prime }/{\rho }_{\ast }$ and $b=-g{\rho }^{\prime }/{\rho }_{\ast }$,This is the Brunt–Väisälä frequency. Its assumed constancy requires a locally uniform stable density gradient in this approximation; rotation and viscosity are neglected.

For a plane wave with wavevector $K=(k,\ell ,m)$, project the momentum equation onto the space perpendicular to $K$ to eliminate pressure. Its vertical component is $-i\omega w=b[1-m^2/|K{|}^2]$, while $-i\omega b+N^{2}w=0$. Eliminating $b$ yieldsFor $0<\omega <N$, a small oscillating body radiates upward and downward conical internal gravity wave beams. In a vertical section the beams make a St Andrew's cross, with angle $\alpha$ to the horizontal satisfying $\sin \alpha =\omega /N$. Indeed the group velocity is perpendicular to the wavevector and has ratio of vertical to horizontal magnitudes $\sqrt{k^2+{\ell }^2}/|m|$. Thus wave energy follows the cones, rather than the direction normal to a fixed-radius spherical wavefront. There are no such propagating beams for $\omega >N$.

For the shear-flow calculation choose the positive intrinsic-frequency branch and $k_0>0$, conditions needed for the printed critical-level formula. The slowly varying local dispersion relation isStationarity and horizontal homogeneity conserve $\Omega$ and $k=k_0$. The Hamiltonian ray equations for a local dispersion relation giveThus $m_0<0$ stays negative and the ray propagates upward. Conservation of frequency givesIf the shear reaches the required velocity difference, its first zero is the critical level of an internal gravity wave,As $z\uparrow z_c$, the intrinsic frequency tends to zero, $m\to -\infty$, the vertical wavelength tends to zero and the vertical group velocity tends to zero. The ideal ray cannot cross this level. For finite $U_c^{\prime }>0$, $\dot{z}\sim k_0(U_c^{\prime }{)}^2(z_c-z{)}^2/N$, so it approaches the level only asymptotically in time. The disturbance develops short scales; even weak viscosity can then absorb it, and nonlinear effects can invalidate the small-amplitude approximation.

The printed sign and existence hypotheses are incomplete. On the positive-frequency branch, $k_0<0$ instead gives $\dot{m}>0$ and eventually a turning point when the intrinsic frequency reaches $N$. As a concrete counterexample to the stated universal height bound, take $N=1$, $k_0=-1$, $m_0=-3$, and $U(z)=\epsilon z$ with any small $\epsilon >0$. At the printed height $z_c=1/(\epsilon \sqrt{10})$, the intrinsic frequency is $2/\sqrt{10}<1$ and $m=-\sqrt{3/2}<0$, so the vertical group velocity is still positive and the ray crosses that height. Also, $U^{\prime }>0$ alone does not ensure that the displayed velocity difference is ever reached; an increasing bounded $U$ may have no finite critical level. The qualified derivation above gives the intended positive-$k_0$ result whenever that level exists.

Use the geometrical optics conventionBecause ${\partial }_t=\epsilon {\partial }_T$ and ${\partial }_x=\epsilon {\partial }_X$, differentiating $e^{i\theta /\epsilon }$ gives an order-one physical angular frequency and order-one physical wavenumbers. Absorb a constant leading amplitude phase into $\theta$ and take $A_0$ real; for a complex amplitude the corresponding quadratic density is $|A_0{|}^2$.

Substitution into the wave equation gives the eikonal equation for a variable-speed wave at leading order:Choose the positive-frequency branch $\omega =\Omega =cK$, $K=\sqrt{k^2+{\ell }^2}$; the negative branch can be treated in exactly the same way. The imaginary first-order terms giveMultiplying by $A_0$ puts this wave-action transport for a variable-speed wave in conservation form:Its density is $\omega A_0^2$ and its flux is that density times the group velocity. Independently, equality of mixed wave phase derivatives givesThis phase compatibility in geometrical optics is essential when converting the differentiated dispersion relation into equations along rays.

For completeness let $\omega =\Omega (\mathbf{k};\mathbf{X},T)$ be a general local dispersion relation. Define ray tracing by $d\mathbf{X}/dT={\mathbf{c}}_g={\nabla }_{\mathbf{k}}\Omega$. Using the chain rule in ${\nabla }_{\mathbf{X}}\omega$ and the symmetry ${\partial }_{X_i}{k}_j={\partial }_{X_j}{k}_i$ givesLikewise ${\omega }_T={\Omega }_T+{\mathbf{c}}_g\cdot {\mathbf{k}}_T$ cancels its advective term because ${\mathbf{k}}_T=-\nabla \omega$. Hence the Hamiltonian ray equations for a local dispersion relation areThe derivatives on the right hold the wavevector fixed. For this wave equation they reduce to

When $c=c(Y)$, the ray invariants are $k$ and $\omega$, while $d\ell /dT=-K{c}_Y<0$. For a monochromatic component with these fixed invariants write $\theta =kX-\omega T+S(Y)$, soUsing the stated independence of $A_0$ from $X,T$, the wave-action transport for a variable-speed wave becomes $(c^2\ell A_0^2{)}_Y=0$. ThereforeThis is the stationary wave amplitude in a stratified wavespeed; $\mu$ is the constant for that component or ray family.

At the proposed simple turning point of a variable-speed wave, take $c_s=c(Y_s)=\omega /k$ and $c_s^{\prime }=c_Y(Y_s)>0$. On the chosen positive branch this entails $k>0$. DefineAs the ray approaches from below, $\ell \sim [q(Y_s-Y){]}^{1/2}$ andThe amplitude diverges and the vertical group velocity $c^2\ell /\omega$ vanishes. The WKB approximation also fails its slow-variation condition $\epsilon |{\ell }_Y|/{\ell }^2\ll 1$ when $Y_s-Y=O({\epsilon }^{2/3})$. The turning-point enhancement of wave amplitude shows that this divergence is a failure of the outer ray approximation, rather than an actual infinite solution.

To derive the inner wavefield directly, separate the conserved horizontal wave phase and frequency in the original differential equation:Dividing by $c^2$ gives ${\epsilon }^2{\Psi }_{YY}+2{\epsilon }^2(c_Y/c){\Psi }_Y+Q(Y)\Psi =0$. Set $Y-Y_s=\delta \eta$. The second derivative is of size ${\epsilon }^2/{\delta }^2$, whereas $Q\sim -q\delta \eta$, so the Airy scaling at a variable-speed wave turning point isThe first-derivative term is smaller by $O(\delta )$, and higher coefficients in the expansion of $Q$ give the same relative order. Thus the general leading inner profile is a linear combination of the two Airy functions $\mathrm{Ai}(\eta )$ and $\mathrm{Bi}(\eta )$. For decaying Airy continuation and ray reflection, if the physical continuation is bounded and decays into $Y>Y_s$, its growing $\mathrm{Bi}$ coefficient is zero. In that usual evanescent continuation the wavefield has the formWithout a condition in the evanescent region the corresponding $\mathrm{Bi}$ term remains admissible locally; the given existence of a wavefield below the turning point alone does not select its coefficient.

Finally, the outer amplitude at distance $\delta$ is $O({\delta }^{-1/4})=O({\epsilon }^{-1/6})$ for $\mu =O(1)$ and fixed nonzero $q$. The oscillatory Airy turning-point connection formula has exactly the same $(-\eta {)}^{-1/4}$ envelope on the allowed side. Therefore the inner wavefield has magnitude $O({\epsilon }^{-1/6})$. The decaying Airy continuation matches an incident and reflected pair of rays, so a single upward $\ell >0$ branch must be supplemented by its reflected branch near the turning point; no formal coefficient matching is needed to identify this magnitude.

A smooth phase gives ${\partial }_{X_i}{k}_j={\partial }_{X_j}{k}_i$ and ${\mathbf{k}}_T=-\nabla \omega$. These mixed-derivative identities allow the spatially differentiated dispersion relation to be rewritten as Hamiltonian ray equations for a local dispersion relation. Away from phase singularities they express the curl-free character of a local wavevector.