Group vs Phase Velocity

Physics: Group velocity v_g = v_p + k·dv_p/dk = dω/dk (energy transport). Phase velocity v_p = ω/k (speed of constant phase). Normal dispersion (dv_p/dk < 0): v_g < v_p (most media). Anomalous dispersion: v_g > v_p or even v_g > c near absorption resonances — but no information travels faster than c. Deep water waves: v_g = ½v_p (crests appear from back of packet, race to front, disappear). Shallow water: non-dispersive, v_g = v_p = √(gh).