Phase velocity

Remarks

The set where the phase $k \cdot x - \omega t + \phi$ has a fixed value is a plane that moves along $k / \abs{k}$ with speed $\omega / \abs{k}$; this is what the definition measures. In a medium with refractive index $n(\omega)$, the dispersion relation $\abs{k} = n(\omega) \omega / c$ gives the phase speed $c / n(\omega)$.

Example: for $u(x, t) = \cos(2x - 6t)$ in one dimension, $k = 2$ and $\omega = 6$, so $v_p = 3$; indeed $u(x + 3s, t + s) = u(x, t)$ for all $s$.

Non-example: the speed of a crest is not $k / \omega$: for the same wave, $k/\omega = 1/3$, and $u(x + s/3, t + s) = \cos(2x - 6t - \tfrac{16}{3} s)$, which changes with $s$.