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Resonance phenomena in a semi‐infinite string with a periodic end
Author(s) -
Werner P.
Publication year - 1993
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.1670160704
Subject(s) - mathematics , countable set , eigenvalues and eigenvectors , mathematical analysis , spectrum (functional analysis) , continuous spectrum , string (physics) , interval (graph theory) , operator (biology) , boundary value problem , periodic boundary conditions , boundary (topology) , pure mathematics , combinatorics , mathematical physics , quantum mechanics , physics , biochemistry , chemistry , repressor , transcription factor , gene
We study the propagation of linear waves, generated by a compactly supported time‐harmonic force distribution, in a semi‐infinite string under the assumption that the material properties depend p ‐period‐ically on the space variable outside a sufficiently large interval [0, a ]. The spectrum of the self‐adjoint extension A of the spatial part of the differential operator consists of a finite or countable number of bands and a (possibly empty) discrete set of eigenvalues located in the gaps of the continuous spectrum. We show that resonances of order t or t ½ , respectively, occur if either ω 2 is an eigenvalue of A or (i) ω 2 is a boundary point of the continuous spectrum of A and (ii) the corresponding time‐independent homogeneous problem has a non‐trivial solution which is p ‐periodic or p ‐semiperiodic for x > a (‘standing wave’).