= Solution
Assume the medium varies on scales much longer than the wavelength and period, so the leading-order <WKB method> gives the local <eikonal equation>. For the phase $\theta$ in the stated ansatz, define
$$
\mathbf k=\nabla\theta,
\qquad
\omega=-\partial_t\theta.
$$
The local <dispersion relation> becomes the Hamilton--Jacobi equation
$$
-\partial_t\theta
=\Omega(\nabla\theta;\mathbf x,t).
$$
Differentiating it with respect to $x_i$, using equality of mixed derivatives, and choosing a ray whose velocity is
$$
\dot x_i=\frac{\partial\Omega}{\partial k_i}
$$
gives
$$
\dot k_i
=\partial_tk_i+\dot x_j\partial_{x_j}k_i
=-\frac{\partial\Omega}{\partial x_i}.
$$
Here the dot, or $d/dt$, is the <total derivative along a ray>,
$$
\frac d{dt}
=\frac{\partial}{\partial t}
+\dot{\mathbf x}\cdot\nabla_{\mathbf x},
$$
and not merely a partial time derivative at fixed position.
Finally, differentiate $\omega=\Omega(\mathbf k;\mathbf x,t)$ along the same trajectory:
$$
\frac{d\omega}{dt}
=\frac{\partial\Omega}{\partial t}
+\frac{\partial\Omega}{\partial x_i}\dot x_i
+\frac{\partial\Omega}{\partial k_i}\dot k_i
=\frac{\partial\Omega}{\partial t},
$$
because the last two terms cancel. The <Hamiltonian ray-tracing equations> are therefore
$$
\boxed{
\frac{dx_i}{dt}=\frac{\partial\Omega}{\partial k_i},
\qquad
\frac{dk_i}{dt}=-\frac{\partial\Omega}{\partial x_i},
\qquad
\frac{d\omega}{dt}=\frac{\partial\Omega}{\partial t}
}.
$$
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