1D periodic mass-spring array with alternating parameters — transmission spectrum and bandgap
A 1D phononic crystal with alternating masses m₁,m₂ and springs k₁,k₂ has a complete bandgap: no elastic wave propagates in the frequency range [ω₋, ω₊]. The gap originates from Bragg scattering at the unit cell boundary (a=a₁+a₂). Band edges: ω²_{1,2} = (k₁+k₂)/m_{1,2}. Transfer matrix method gives exact transmission T(ω) = 4/(4+|Tr(M)²−4|) per unit cell. Inside the gap T decays exponentially with N.