Transmission of acoustic waves in a random layered medium

The transmission of acoustic waves through a sequence of alternating layers with random thicknesses but otherwise fixed characteristics is studied by means of the transfer-matrix formalism of one-dimensional disordered chains. The law ${\lim }_{\mathrm{N}\to \mathrm{\infty }}$ln(‖$T_N$‖/N)≡-λ(ω) of the exponential decay of the transmission coefficient $T_N$ as a function of the number (2N) of layers is determined in a weak- (strong-) disorder regime for an arbitrary (uniform) distribution of layer thicknesses. The localization constant λ(ω) has a particularly simple form at extreme low and high frequencies ω. Namely λ(ω→0)=const×${\omega }^2$ with a slope given in terms of physical characteristics of the layers and λ(ω→∞)=const defined by a transmission coefficient of a single interface. The predictions are tested by Monte Carlo simulations of a simple model with characteristics of certain rocks. For all frequencies beyond the weak-strong disorder turnover region discrepancies between theoretical and numerical results are merely a few percent.