Localization and mode conversion for elastic waves in randomly layered media I

This paper is Part I of a two-part work in which we derive localization theory for elastic waves in plane-stratified media, a multimode problem complicated by the interconversion of shear and compressional waves, both in propagation and in backscatter. We consider the low frequency limit, i.e., when the randomness constitutes a microstructure. In this part, we set up the general suite of problems and derive the probability density and moments for the fraction of reflected energy which remains in the same mode (shear or compressional) as the incident field. Our main mathematical tool is a limit theorem for stochastic differential equations with a small parameter. In Part II we will use the limit theorem of Part I and the Oseledec Theorem, which establishes the existence of the localization length and other structural information, to compute: the localization length and another deterministic length, called the equilibration length, which gives the scale for equilibration of shear and compressional energy in propagation; 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 field. We also extend the results to the small fluctuation, rather than the low frequency case.