Group velocity

Remarks

A wave packet $u(x, t) = \int a(k) e^{i(k \cdot x - W(k) t)} dk$ with an amplitude $a$ concentrated near $k_0$ moves, to first order, with velocity $v_g(k_0)$: expanding $W(k) \approx W(k_0) + v_g(k_0) \cdot (k - k_0)$ gives $u(x, t) \approx e^{i(k_0 \cdot x - W(k_0) t)} \, f(x - v_g(k_0) t)$ for a fixed envelope $f$. In one dimension, $v_g = v_p + k \, dv_p / dk$, where $v_p = \omega / k$ is the phase velocity; the two agree exactly when the equation is non-dispersive.

Example: light in a cold plasma has $\omega = \sqrt{\omega_p^2 + c^2 k^2}$ with plasma frequency $\omega_p > 0$. Then $v_g = c^2 k / \omega < c$ and $v_p = \omega / k > c$, and $v_g v_p = c^2$.

Non-example: the ratio $\omega / k$ is not the group velocity in a dispersive medium. For deep water waves, $\omega = \sqrt{g k}$ with gravitational acceleration $g$, so $v_g = \frac{1}{2} \sqrt{g / k}$, while $\omega / k = \sqrt{g / k}$.