= Solution
For the rapidly varying <wave phase>, define $k_i=\partial_i\theta$ and $\omega=-\partial_t\theta$, with the common scaling by $\varepsilon$ absorbed into these local variables. A packet moves with <group velocity> $c_{g,i}=\partial\Omega/\partial k_i$. Along that trajectory, $d/dt=\partial_t+c_g\cdot\nabla_x$. Compatibility gives $\partial_tk_i=-\partial_i\omega$ and $\partial_i k_j=\partial_j k_i$. Differentiating the local <dispersion relation> then cancels its implicit $k$ <derivatives>, yielding
$$
\boxed{\dot x_i=\Omega_{k_i},\quad\dot k_i=-\Omega_{x_i},\quad\dot\omega=\Omega_t.}
$$
The last equality follows also by differentiating $\omega=\Omega(k,x,t)$ along the ray and cancelling the two <Hamiltonian> cross terms. The partial <derivatives> of $\Omega$ hold its other arguments fixed.
A mean flow adds the <Doppler shift>: the <intrinsic frequency> is $\widehat\omega=\omega-kU(z)$, so
$$
(\omega-kU)^2=\frac{N^2k^2}{k^2+m^2},\qquad
\Omega=kU+\frac{Nk}{\sqrt{k^2+m^2}}
$$
on the positive-intrinsic-frequency branch launched here. The medium is stationary and independent of $x$, so $\omega$ and $k$ are constant. Initially $m=0,U=0$, hence $\omega=N$. Moreover $\dot m=-kU'<0$, and then $\dot z=-Nkm/(k^2+m^2)^{3/2}>0$. The initial vertical <group velocity> is zero, but $m$ immediately becomes negative and the ray accelerates upwards.
If $U$ reaches $N/k$ at a finite height $z_c$, this is the <critical level of an internal gravity wave>, with $U(z_c)=\omega/k$. Below it,
$$
\boxed{m=-k\sqrt{\frac{N^2}{(N-kU)^2}-1}.}
$$
Assuming $0<U'(z_c)<\infty$, put $\delta=z_c-z$. Then $N-kU\sim kU'(z_c)\delta$, giving
$$
\boxed{m\sim-\frac{N}{U'(z_c)\delta},\quad
\dot z\sim\frac{kU'(z_c)^2}{N}\delta^2,\quad
t\sim\frac{N}{kU'(z_c)^2\delta}\longrightarrow\infty.}
$$
Thus the ray cannot cross this level in finite time within the geometrical-optics approximation. The hypothesis $U'>0,U(0)=0$ alone does not guarantee a finite $z_c$: for example $U(z)=\frac{N}{2k}(1-e^{-z})$ on $z\geq0$ never reaches $N/k$. \b[Existence of a finite critical level is an additional assumption]. Very close to one, diverging <wavenumber> can invalidate the inviscid, linear, slowly varying approximation; the result is the formal ray prediction.
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