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Resonances in periodic media
Author(s) -
Werner P.
Publication year - 1991
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670140403
Subject(s) - mathematical analysis , mathematics , bounded function , wavenumber , resonance (particle physics) , operator (biology) , transverse plane , distribution (mathematics) , boundary (topology) , cylinder , boundary value problem , harmonic , physics , geometry , acoustics , optics , quantum mechanics , biochemistry , chemistry , structural engineering , repressor , transcription factor , gene , engineering
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).