Phase compatibility in geometrical optics
= Phase compatibility in geometrical optics
{title2=$k_Y=\ell_X,\quad\mathbf k_T=-\nabla\omega$}
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.