ASYMPTOTIC ANALYSIS OF WAVE PROPAGATION ALONG WEAKLY NON-UNIFORM REPETITIVE SYSTEMS

We consider monochromatic wave propagation along a long, finite, one-dimensional, slightly non-uniform waveguide, whose ends are connected to uniform semi-infinite waveguides. The non-uniformity in the system parameters, which is assumed slowly varying and deterministic, can be tuned to produce a desired scattered wave field or reflection/transmission properties for a broad 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 systems governed by a second order finite difference equation and apply the WKB method allowing the index variable to be complex. This allows complex turning points to be considered. The coefficients of the difference equation are represented by their discrete Fourier modes. For complex turning points, we obtain exponentially small reflection, a new result in the context of difference equations. The asymptotic solution, besides revealing how the non-uniformity in the parameters affects wave propagation, furnishes an analytic 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 results with numerical experiments for large finite one-dimensional non-uniform chains of coupled pendula.