Solution
= Solution
For the given <internal gravity wave>,
$$
\Omega(k,m;x,z)
=\frac{Nk}{\sqrt{k^2+m^2}}
+\gamma(kx-mz).
$$
The wavevector part of the <Hamiltonian ray-tracing equations> is
$$
\dot k=-\frac{\partial\Omega}{\partial x}=-\gamma k,
\qquad
\dot m=-\frac{\partial\Omega}{\partial z}=\gamma m.
$$
Applying the initial data gives
$$
\boxed{
k(t)=k_0e^{-\gamma t},
\qquad
m(t)=m_0e^{\gamma t}
}.
$$
In particular, $k(t)m(t)=k_0m_0$ is conserved.