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 ω² is an eigenvalue of A or (i) ω² 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’).