Abstract
This paper presents a probabilistic study of the effects of structural irregularity on wave propagation along an infinite 1-D chain. A general integral equation method based on Markov chain theory is used to determine the phase probability density function (pdf) at the scatterers distributed irregularly along the chain. The scatterers could be atoms in a one-dimensional crystal, or ribs on a flat plate or membrane. The integral equation derived for the phase pdf is simplified considerably when the scatterers are distributed completely randomly or quasi-periodically. In these cases, the integral equations may be asymptotically solved for the phase density functions in the limit of weak or strong scattering; the localization factors are then obtained. The present approach is quite general and is directly applicable to any disordered one-dimensional system consisting of identical scatterers that are arranged according to a probability distribution function. The validity of the present asymptotic solutions is examined and verified by comparing against the existing analytical solutions for simple atomic or mechanical disordered systems.
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