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

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’).