Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 39C / a

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, defineThe local dispersion relation becomes the Hamilton--Jacobi equationDifferentiating it with respect to $x_i$, using equality of mixed derivatives, and choosing a ray whose velocity isgivesHere the dot, or $d/dt$, is the total derivative along a ray,and not merely a partial time derivative at fixed position.

Finally, differentiate $\omega =\Omega (\mathbf{k};\mathbf{x},t)$ along the same trajectory:because the last two terms cancel. The Hamiltonian ray-tracing equations are therefore