Local equality of phase and group velocity ($v_g-v_p=kd{v}_p/dk$)

For a differentiable dispersion relation and nonzero wavenumber $k$, group velocity equals phase velocity precisely where $d{v}_p/dk=0$. Equality at an isolated stationary point of $v_p$ does not establish a nondispersive branch. For a gravity-capillary wave, the equality occurs at $k=\sqrt{\rho g/T}$ and the phase-speed minimum, even though the relation is dispersive elsewhere.