A 1D periodic array of alternating materials creates Bragg bandgaps where elastic waves cannot propagate. The gap opens at k = π/a (Brillouin zone boundary) due to destructive interference.
Transfer matrix method: at each interface, the transfer matrix M = [[cos(kd), sin(kd)/Z], [-Z sin(kd), cos(kd)]] connects fields across layers. The band structure follows from Floquet-Bloch: cos(Ka) = ½Tr(M_cell). Bandgap opens when |½Tr| > 1. Gap width ∝ impedance contrast.