Asymptotic analysis of wave propagation along weakly non-uniform repetitive systems

We consider monochromatic wave propagation along a long, finite, one-dimens ional, slightly non-uniform waveguide, whose ends are connected to uniform semi-infinite waveguides. The non-uniformity in the system parameters, whic h is assumed slowly varying and deterministic, can be tuned to produce a de sired scattered wave field or reflection/transmission properties for a broa d range of incident wave fields. With this objective in mind, we obtain an analytic solution for wave propagation along repetitive systems, asymptotic in the slowness of the variation of the system parameters. We consider sys tems governed by a second order finite difference equation and apply the WK B method allowing the index variable to be complex. This allows complex tur ning points to be considered. The coefficients of the difference equation a re represented by their discrete Fourier modes. For complex turning points, we obtain exponentially small reflection, a new result in the context of d ifference equations. The asymptotic solution, besides revealing how the non -uniformity in the parameters affects wave propagation, furnishes an analyt ic expression for the system scattering matrix as a function of the system parameters. It also sheds light on the mechanism of localization phenomena for this class of repetitive systems. We also compare the asymptotic result s with numerical experiments for large finite one-dimensional non-uniform c hains of coupled pendula. (C) 2000 Academic Press.