Past exam of the mathematical methods II course of the University of Oxford / 2013 / 10 / e / Solution

A local-wave description requires an identifiable smooth phase $\Theta (x,t)$ with $\bar{k}={\Theta }_x$ and $\bar{\omega }=-{\Theta }_t$. Real propagating wavenumber and frequency must satisfy the chosen local branch of ${\bar{\omega }}^2=c^2{\bar{k}}^2+a$. In a geometrical optics or slowly modulated wavepacket approximation, these quantities and the amplitude vary on scales large compared with a local wavelength and period; for example $|{\partial }_x\bar{k}|\ll {\bar{k}}^2$ and $|{\partial }_t\bar{\omega }|\ll {\bar{\omega }}^2$ away from degenerate points.

The phase compatibility in geometrical optics means the local fields cannot be specified independently: equality of mixed phase derivatives requires ${\bar{k}}_t+{\bar{\omega }}_x=0$, derived as wave conservation below. These phase and scale conditions justify the local dispersion relation; they alone are not a guarantee of an arbitrary-amplitude exact PDE solution. Amplitude transport and higher-order envelope terms must also be satisfied.