Magnon dispersion (1D, nearest-neighbor Heisenberg): ω(k) = 2JS(1 − cos ka) + 2KS, where J is exchange coupling, K is easy-axis anisotropy, S is spin magnitude, and a is lattice spacing. At small k: ω ≈ JSk²a² (quadratic, unlike phonons which are linear). The group velocity v_g = dω/dk is the speed of energy transport; phase velocity v_p = ω/k differs, causing dispersion. Upper canvas shows the full dispersion curve; lower canvas shows individual spins precessing in the selected spin-wave mode.