Localization and mode conversion for elastic waves in randomly layered media

We derive localization theory for elastic waves in plane-strati ed media, a multimode problem complicated by the interconversion of shear and compressional waves, both in propagation and in backscatter. In the low frequency limit, i.e. when the randomness constitutes a microstructure, we give analytical expressions for the following quantities: the localization length, and another deterministic length, called the equilibration length, which gives the scale for the equilibration of compressional and shear energy in propagation; the probability density of the fraction of re ected energy which remains in the same mode (shear or compressional) as the incident eld; and the probability density of the ratio of shear to compressional energy in transmission through a large slab. This last quantity is shown to be asymptotically independent of the incident eld. Our main mathematical tools are: the Oseledec theorem, which establishes the existence of the localization length and other structural information, and limit theorems for stochastic di erential equations with a small parameter.