Abstract
Two distinct asymptotic theories for wave propagation in one-dimensional inhomogeneous media are compared in their common domain of validity. One theory, due to Santosa and Symes, applies to long wavelength propagation in periodic media with arbitrary contrast in material properties. The O’Doherty–Anstey theory, on the other hand, is explicitly intended to describe time-dependent wave propagation in media that are finely layered but characterized by relatively small reflectivity. The two theories are compared in detail in the doubly asymptotic limit of low-frequency wave propagation in periodic media with small contrasts. The equivalence is demonstrated by deriving the asymptotic limit of the dispersion curve of the fundamental Bloch wave according to each theory. The analysis for the O’Doherty–Anstey theory sheds some new light on its strengths and limitations, particularly in periodic media. It is shown that it correctly predicts the leading-order dispersion curve of the first branch for frequencies of $O( 1 )$, but fails near the first band edge.
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