= Solution
Take a common constant <buoyancy frequency> $N$ and put $k=k_T>0$. Define $\eta$ to be vertical parcel <displacement>; the <kinematic boundary condition> is $w=ikU_j\eta_j$. In either uniform region the exact <Taylor–Goldstein equation> gives
$$
\eta_j''+m_j^2\eta_j=0,\qquad m_j^2=\frac{N^2}{U_j^2}-k^2.
$$
No <WKB approximation> is needed. If the earlier $N(z)$ is genuinely nonconstant, the exact regional equation is instead $\eta_j''+[N(z)^2/U_j^2-k^2]\eta_j=0$; its fundamental solutions, not plane exponentials, must be used. For example, let $C_1(0)=1$, $C_1'(0)=0$, $S_1(0)=0$, $S_1'(0)=1$, and let $f_2$ be an outgoing upper solution. Then $\eta_1=h_0C_1+D S_1$, $\eta_2=A f_2$, with $D,A$ determined by $h_0C_1(H)+DS_1(H)=Af_2(H)$ and $U_1^2[h_0C_1'(H)+DS_1'(H)]=U_2^2Af_2'(H)$. This gives the exact arbitrary-profile field implicitly, while a simple angle-only coefficient requires further stratification assumptions.
Under the common constant-$N$ assumption, propagating incident <internal gravity waves> require
$$
\boxed{0<U_1<\frac Nk.}
$$
For co-directed upper flow, the outgoing propagating branch exists for $0<U_2<N/k$ and has $m_2>0$. The <radiation condition> chooses $m_j>0$ because the laboratory vertical <group velocity> is positive on the negative <intrinsic frequency> branch. Define $\tan\theta_j=m_j/k$, so $\cos\theta_j=kU_j/N$.
At the <velocity> jump, the material-interface <jump conditions for stratified inviscid shear flow> require continuity of <displacement> and of <fluid pressure>. Vertical <velocity> itself is generally discontinuous: $w_j=ikU_j\eta$. In terms of <displacement> the pressure is $p_j=U_j^2\eta_j'$. The <displacement impedance for an internal wave at a velocity jump> is $Z_j=U_j^2m_j$. An upward incident component of complex <displacement> <amplitude> $I$ at $H$, a reflected component $RI$ and a transmitted component $TI$ therefore obey
$$
1+R=T,\qquad Z_1(1-R)=Z_2T.
$$
Consequently
$$
\boxed{R=\frac{Z_1-Z_2}{Z_1+Z_2},\qquad T=\frac{2Z_1}{Z_1+Z_2}.}
$$
For a one-pass incident <displacement> $h_0$, a real wave field is the imaginary part of
$$
\begin{aligned}
\eta_1(x,z)&=h_0e^{ikx}\left[e^{im_1z}+R e^{im_1(2H-z)}\right],&&0<z<H,\\
\eta_2(x,z)&=h_0T e^{i[kx+m_1H+m_2(z-H)]},&&z>H.
\end{aligned}
$$
The incident vertical-<displacement> <amplitude> is $h_0$, the reflected <amplitude> is $|R|h_0$ and the transmitted <amplitude> is $|T|h_0$; the sign of $R$ encodes a phase reversal. The other fields follow from $w=ikU\eta$, $u=-w'/(ik)$, $b=-N^2\eta$ and $p=U^2\eta'$. If a perfectly reflecting terrain boundary is enforced after the return wave arrives, replace the incident coefficient $h_0$ by $h_0/[1+Re^{2im_1H}]$. The one-pass <amplitudes> use the permitted neglect of repeated terrain reflections.
For common $N$, $Z_j=(N^2/k)\cos\theta_j\sin\theta_j$. The physical vertical-<displacement> transmission coefficient is therefore
$$
\boxed{\frac{\eta_{2,\mathrm{vertical}}}{h_0}=\frac{2\cos\theta_1\sin\theta_1}{\cos\theta_1\sin\theta_1+\cos\theta_2\sin\theta_2}=\frac{2\cos\theta_1\sin\theta_1}{\sin(\theta_1+\theta_2)\cos(\theta_1-\theta_2)}.}
$$
The printed coefficient has a different denominator and cannot be obtained for this standard <displacement> convention. For example, when $U_2=U_1$, there is no interface scattering and the vertical <amplitude> stays $h_0$, whereas the printed expression gives $h_0/\cos\theta_1$. A total <displacement> magnitude would instead be the vertical <amplitude> divided by $\cos\theta_2$; with unequal <velocities> it still does not give the printed expression. \b[The transmission formula needs an explicit correction or a different, presently unspecified <amplitude>/interface model.]
For $U_2>N/k$, choose $m_2=i\kappa$ with $\kappa=(k^2-N^2/U_2^2)^{1/2}$: the transmitted response decays as $e^{-\kappa(z-H)}$ and the same impedance equations give a complex $R,T$, with $|R|=1$. At $U_2=0$ the steady equation is singular. Oppositely directed upper flow formally supports propagating waves when $0<|U_2|<N/k$, but upward radiation then requires $m_2<0$ and signed $Z_2$; the positive-angle co-directed formula is not valid there. A real <velocity> jump can also have <Kelvin-Helmholtz instability>, which is excluded from this formal steady scattering calculation.
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