Statistics of the Lyapunov exponent in one-dimensional layered systems

Localization of acoustic waves in a one-dimensional water duct containing many randomly distributed air filled blocks is studied. Both the Lyapunov exponent and its variance are computed. Their statistical properties are also explored extensively. The results reveal that in this system a single-parameter scaling is generally inadequate no matter whether the frequency we consider is located in a pass band or in a band gap. This contradicts the earlier observations in an optical case. We compare the results with two optical cases and give a possible explanation of the origin of the different behaviors.