= Solution
The <WKB approximation> treats the vertical <wavenumber> as locally constant over one vertical <wavelength>. Write $w=\Re\{A(z)e^{ikx+i\int^z m(s)ds}\}$, with $|m'|\ll m^2$ and $|A'|\ll |mA|$. For the <stationary internal-wave WKB solution>, $A\propto m^{-1/2}$ away from singular or turning levels. A small terrain slope and small perturbations are also needed for <Linearized Boussinesq equations>.
Let $p$ be the perturbation <fluid pressure> divided by reference <mass density>, and $b$ the <buoyancy perturbation>. The steady <Linearized Boussinesq equations> and <incompressibility condition> are
$$
Uu_x+U'w=-p_x,\qquad Uw_x=-p_z+b,\qquad Ub_x+N^2w=0,\qquad u_x+w_z=0.
$$
For an $e^{ikx}$ component, elimination gives $u=-w'/(ik)$, $b=-N^2w/(ikU)$ and $p=(Uw'-U'w)/(ik)$. Substitution into vertical momentum gives the exact stationary <Taylor–Goldstein equation>
$$
w''+\left(\frac{N^2}{U^2}-\frac{U''}{U}-k^2\right)w=0.
$$
The leading <WKB approximation> therefore identifies
$$
\boxed{m^2=\frac{N^2}{U^2}-\frac{U''}{U}-k^2=\left(\frac{N^2}{k^2U^2}-\frac{U''}{k^2U}-1\right)k^2.}
$$
The <kinematic boundary condition> fixes the horizontal <wavenumber> to $k=k_T$ for the sinusoidal forcing.
For explicit heights assume a constant positive <buoyancy frequency> $N$, $0<kU_0<N$, and $|S|/N\ll1$. The latter is a large <gradient Richardson number>. The <linear shear flow> has $U''=0$, so the propagation condition is $|U|<N/k$.
For $S<0$, the first obstruction is the <critical level of an internal gravity wave> at
$$
\boxed{z_c=\frac{U_0}{|S|}.}
$$
As $z\uparrow z_c$, $m\sim N/U\to\infty$. The <critical-height approach of a stationary wave in linear shear> has upward <group velocity> $c_{gz}\sim kU^2/N$, so the travel time to $z_c$ diverges: the height is a supremum, approached in the ideal ray model. Here $|m'|/m^2\to |S|/N$, which can remain small; it would be incorrect to say that the short-<wavelength> <WKB approximation> necessarily fails solely because $m$ diverges. Nevertheless $w\propto U^{1/2}$ and <displacement> $w/(ikU)\propto U^{-1/2}$, so a finite disturbance eventually violates linearity. Dissipation and overturning also matter near this singular level.
For $S>0$, the <turning height of a stationary wave in linear shear> is
$$
\boxed{z_t=\frac{N/k-U_0}{S}.}
$$
At $z_t$, $m=0$ and the <WKB approximation> fails. An <Airy function> matching region describes reflection and an upper <evanescent wave>. Thus $z_t$ is the highest height of vertical propagation, rather than a sharp cut-off of every disturbance.
The problem initially allows $N(z)$ to vary. Without the extra constant-$N$ assumption, replace the explicit turning height by the first root of $N(z)=k|U_0+Sz|$ encountered on a propagating branch. A declining $N(z)$ can cause a turning level before the negative-shear critical level. \b[The two closed-form heights require the stated constant-stratification assumption.]
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