Wave Packets are superpositions of plane waves: ψ(x,t) = ∫ dk φ(k) e^{i(kx−ω(k)t)}. The shape of the dispersion relation ω(k) determines how the packet propagates and spreads.
v_phase = ω/k | v_group = dω/dk | GVD = d²ω/dk² (spreading rate)
Non-dispersive media (v_group = v_phase = const) preserve packet shape. Dispersive media cause spreading: Δx(t) ≈ √(Δx₀² + (GVD·t/Δx₀)²). A quantum free particle always spreads: σ(t) = σ₀√(1+(ℏt/mσ₀²)²). Deep-water waves have v_group = v_phase/2 — energy travels at half the wave crest speed.