LOCALIZATION AND MODE CONVERSION FOR ELASTIC-WAVES IN RANDOMLY LAYERED MEDIA .1.

This paper is Part I of a two-part work in which we derive localizatio n theory for elastic waves in plane-stratified media, a multimode prob lem complicated by the interconversion of shear and compressional wave s, both in propagation and in backscatter. We consider the low frequen cy limit, i.e., when the randomness constitutes a microstructure. In t his part, we set up the general suite of problems and derive the proba bility density and moments for the fraction of reflected energy which remains in the same mode (shear or compressional) as the incident fiel d. Our main mathematical tool is a limit theorem for stochastic differ ential equations with a small parameter. In Part II we will use the li mit 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 equilibrat ion of shear and compressional energy in propagation; and the probabil ity density of the ratio of shear to compressional energy in transmiss ion through a large slab. This last quantity is shown to be asymptotic ally independent of the incident field. We also extend the results to the small fluctuation, rather than the low frequency case.