A 2D square lattice of atoms coupled by springs. The phonon dispersion relation ω(k) = 2√(K/m)|sin(k_x/2)|... shows how wave speed depends on wavelength. Visualize real-space atomic motion and the dispersion surface over the Brillouin zone.
Dispersion: ω² = (4K₁/m)[sin²(k_x/2)+sin²(k_y/2)] + (8K₂/m)[sin²((k_x+k_y)/2)+sin²((k_x−k_y)/2)] | BZ corners: (π,π) is the zone boundary