Resonances in periodic media

We study the propagation of linear acoustic waves (a) in an infinite string with a periodic material distribution, (b) in an infinite cylinder with a meterial distribution that is periodic in the longitudinal direction and does not depend on the transverse coordinates. We assume that the wave field is generated by a time‐harmonic force distribution of frequency ω acting in a compact set. We show in both cases that resonances of order t 1/2 occur for a discrete set of frequencies and that the solution is bounded as t →∞ for the remaining frequencies. In case (a) ω is a resonance frequency if and only if ω 2 is a boundary point of one of the spectral bands of the corresponding spatial differential operator of Hill's type. A similar characterization of the resonance frequencies is given in case (b).