1D diatomic phononic crystal: Two masses (m₁, m₂) with spring constants k₁, k₂ per unit cell. Bloch's theorem gives dispersion: cos(kα) = cos(φ₁)cos(φ₂) − ½(r + 1/r)sin(φ₁)sin(φ₂) where r = √(k₂m₁/k₁m₂), φᵢ = ω√(mᵢ/kᵢ)α/2.
Band folding: the Brillouin zone boundary at k = π/α creates acoustic and optical branches. The stop band (band gap) opens at k = π/α with gap ∝ |m₂−m₁|.
Phononic crystals control acoustic wave propagation analogously to photonic crystals for light.