A phononic crystal is a periodic arrangement of materials with contrasting elastic properties. Destructive interference of acoustic waves creates forbidden frequency bands — phononic band gaps — analogous to electronic band gaps in semiconductors.
Phononic Crystal Structure
Dispersion Relation & Transmission
Band gap condition: For a 1D bilayer crystal with layer thicknesses d₁, d₂ and impedances Z₁, Z₂, band gaps open at k = nπ/a. Gap width: Δω/ω₀ ≈ (2/π)|ln(Z₁/Z₂)|·(f·(1−f))^(1/2). Wide gaps from high impedance contrast and f ≈ 0.5.