= Solution
The maximum boundary slope is $s_{\max}=h_0k_T$, so <subcritical internal-wave reflection> requires
$$
h_0k_T<\frac{|\omega|}{\sqrt{N^2-\omega^2}},
$$
or equivalently
$$
\boxed{
\frac{Nh_0k_T}{\sqrt{1+h_0^2k_T^2}}<|\omega|<N
}.
$$
Take $k,m>0$ without loss of generality, put
$$
\mu=\frac mk=\sqrt{\frac{N^2}{\omega^2}-1},
\qquad
p_j=k+jk_T,
$$
and write the incident field as the imaginary part of $\widetilde w_i e^{i(kx+mz-\omega t)}$. The <kinematic boundary condition> on $z=h(x)$ is
$$
[w-u h_x]_{z=h(x)}=0.
$$
For a flat boundary the reflected wave is $-\widetilde w_i\sin(kx-mz-\omega t)$. Expanding the boundary condition in a <Taylor expansion> about $z=0$ creates the <topographic sidebands of an internal gravity wave>. With upward-radiating vertical wavenumber $-\mu|p_j|$, their complex amplitudes through second order are
$$
B_0=-\widetilde w_i
+\frac{\mu^2h_0^2k}{2}
\left(p_1+|p_{-1}|\right)\widetilde w_i,
$$
$$
B_1=-\mu h_0p_1\widetilde w_i,
\qquad
B_{-1}=\mu h_0p_{-1}\widetilde w_i,
$$
$$
B_2=-\frac12\mu^2h_0^2p_1p_2\widetilde w_i,
\qquad
B_{-2}=-\frac12\mu^2h_0^2|p_{-1}|p_{-2}\widetilde w_i.
$$
Thus the general compact result is
$$
w_r(x,z,t)
=\operatorname{Im}\left\{
e^{-i\omega t}
\sum_{j=-2}^{2}B_j
e^{ip_jx-i\mu|p_j|z}
\right\}+O(h_0^3).
$$
For the convenient nondegenerate case $k>2k_T$, all displayed sideband wavenumbers are positive. Defining $\Theta_j=p_jx-\mu p_jz-\omega t$, the same answer is the explicitly real formula
$$
\begin{aligned}
\frac{w_r}{\widetilde w_i}
={}&\left(-1+\mu^2h_0^2k^2\right)\sin\Theta_0\\
&+\mu h_0\left(p_{-1}\sin\Theta_{-1}-p_1\sin\Theta_1\right)\\
&-\frac{\mu^2h_0^2}{2}
\left(p_{-1}p_{-2}\sin\Theta_{-2}+p_1p_2\sin\Theta_2\right)
+O(h_0^3).
\end{aligned}
$$
Validity requires a linear incident wave, an inviscid uniformly stratified bulk, an outgoing-radiation condition, strict separation from critical slopes, and small boundary excursions for every retained mode, in particular $h_0k_T\ll1$ and $\mu h_0\max_{|j|\leq2}|p_j|\ll1$. A vanishing $p_j$ is a degenerate zero-horizontal-wavenumber case and must be treated by taking the corresponding zero-amplitude limit rather than dividing by $p_j$.
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