Multimode, one-dimensional wave propagation in a highly discontinuous medium

Abstract

A pulse propagates obliquely through a one-dimensional medium consisting of a large number N of homogeneous elastic layers. As it propagates, the directly transmitted principal arrival is reduced by transmission loss at each interface, but close to this arrival is a broad pulse, made up of multiply scattered energy, which ultimately appears to diffuse about a moving center. This may be regarded as an extension to multimode propagation of the phenomenon first studied by O'Doherty and Anstey in 1971 and further corroborated and elucidated by many authors since then. When the reflection coefficients at the interface are scaled as 1/ √ N while N → ∞, and when time is measured in units of travel time across an average layer, the shape of the broad pulse approaches a limiting form. which propagates according to an integrodifferential equation analogous to the Kolmogorov-Feller forward equation in probability theory.