= 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.
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