= Solution
Assume the condition in part (ii), so $m_0<0$. Substituting the wavevector solution into the vertical ray equation gives the linear <ordinary differential equation>
$$
\dot z+\gamma z=F(t),
$$
where
$$
F(t)
=\frac{-Nk_0m_0}
{\left(k_0^2e^{-2\gamma t}+m_0^2e^{2\gamma t}\right)^{3/2}}
>0.
$$
The <integrating factor> $e^{\gamma t}$ gives
$$
z(t)
=e^{-\gamma t}
\left[
z_0+\int_0^te^{\gamma s}F(s)\,ds
\right].
$$
The bracket is strictly positive, so $z(t)>0$ at every finite time. Moreover,
$$
F(t)=O(e^{-3\gamma t}),
\qquad
e^{\gamma t}F(t)=O(e^{-2\gamma t}),
$$
so the improper integral converges to a finite positive value. It follows that
$$
\boxed{z(t)\sim Ce^{-\gamma t}\longrightarrow0
\quad(t\to\infty)}
$$
for some $C>0$. The packet therefore approaches $z=0$ asymptotically and never reaches it at finite time. This is the <vertical trapping of an internal gravity wave by planar strain>.
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