 Eikonal equation for a variable-speed wave

For the wave equation ${\phi }_{tt}-\nabla \cdot (c^2\nabla \phi )=0$ with slow variables and phase $e^{i\theta /\epsilon }$, define $\omega =-{\theta }_T$ and $\mathbf{k}=\nabla \theta$. The leading balance is ${\omega }^2=c^2|\mathbf{k}{|}^2$. Its positive-frequency dispersion relation is $\Omega =c|\mathbf{k}|$, and the group velocity is $c\mathbf{k}/|\mathbf{k}|$.