Group vs Phase Velocity in a Dispersive Medium

What you are seeing: a real wave packet, a Gaussian band of wavenumbers added together, travelling through a dispersive medium. This is not a two-tone beat: the carrier crests (gold) move at the phase velocity $v_p = \omega/k_0$, the envelope (cyan) moves at the group velocity $v_g = \mathrm{d}\omega/\mathrm{d}k$, and the two trackers separate when they differ. When $v_p \ne v_g$ you can watch crests being born at one edge of the packet and dying at the other as it moves. And because $\omega(k)$ is curved, the components run at slightly different speeds and the packet spreads as it goes, the hallmark of dispersion that a fixed beat pattern can never show. The lower panel is the dispersion curve $\omega(k)$: the phase velocity is the slope of the chord from the origin, the group velocity is the slope of the tangent at $k_0$, so a straight relation gives $v_p = v_g$ (light) and a curved one makes them differ (deep water $v_g = v_p/2$, Schrödinger $v_g = 2v_p$, an anomalous branch where they point opposite ways).