= Wave conservation equation
{title2=$k_t+\omega_x=0$}
For a smooth phase $\Theta$, define local <wavenumber> $k=\Theta_x$ and local <angular frequency> $\omega=-\Theta_t$. Equality of mixed derivatives gives $k_t+\omega_x=0$. Dividing by $2\pi$ expresses conservation of signed crest count with density $k/(2\pi)$ and flux $\omega/(2\pi)$. When a homogeneous local <dispersion relation> gives $\omega=\omega(k)$, the equation becomes $k_t+v_g(k)k_x=0$; its <characteristic curves> move at the <group velocity>. This phase compatibility is distinct from energy or wave-action conservation.
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