N masses on an infinite string and related one-dimensional scattering problems

One-dimensional time-harmonic waves interact with a finite number of scatterers: they could be beads on a long string, for example. If the scatterers are identical and equally spaced, such periodic problems can be solved exactly. One problem solved here arises when one scatterer in a periodic row is forced to oscillate, giving the Green function for the row. Our main interest is with disordered problems, where a periodic configuration is disturbed. Two problems are studied. First, just one scatterer in a finite periodic row is displaced: an exact solution is obtained for the transmission coefficient and its average over all allowable displacements. Second, a similar problem is treated where each scatterer is displaced by a small distance from its position in the periodic row. The main tools used are perturbation theory and transfer matrices.