Dispersion relation and band gap from periodic layered media
A periodic stack of alternating layers (impedances Z₁,Z₂, thicknesses d₁,d₂, speeds c₁,c₂) creates a phononic crystal. The Bloch wave dispersion satisfies the transfer matrix condition:
When |rhs| > 1, no real K exists — this is the Bragg stopband (band gap). Waves are exponentially attenuated. The gap opens near ω = nπc/d (Bragg condition λ = 2a). Higher impedance contrast → wider gaps.