Phase compatibility in geometrical optics ($k_Y={\ell }_X,{\mathbf{k}}_T=-\nabla \omega$)

A smooth phase gives ${\partial }_{X_i}{k}_j={\partial }_{X_j}{k}_i$ and ${\mathbf{k}}_T=-\nabla \omega$. These mixed-derivative identities allow the spatially differentiated dispersion relation to be rewritten as Hamiltonian ray equations for a local dispersion relation. Away from phase singularities they express the curl-free character of a local wavevector.

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.