Resonance phenomena in a semi‐infinite string with a periodic end

摘要

AbstractWe 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 dependp‐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 ordertort½, respectively, occur if either ω2is an eigenvalue of A or (i) ω2is a boundary point of the continuous spectrum of A and (ii) the corresponding time‐independent homogeneous problem has a non‐trivial solution which isp‐periodic orp‐semiperiodic for