Wave propagation with tunneling in a highly discontinuous layered medium

Abstract

An impulsive plane wave traverses a stratified medium consisting of a large number N of homogeneous isotropic perfectly elastic layers. The directly transmitted wave is greatly reduced by the cumulative effect of scattering loss at each of the many interfaces. However, close to the arrival of the direct wave is a broad pulse, arising from multiple scattering; this pulse does not decay as rapidly as the direct wave and ultimately appears to diffuse about a moving center. The latter process, which is determined by the medium statistics, leads to time delays, effective anisotropy, and apparent attenuation. The present work may be regarded as an extension of that described by Burridge, White and Papanicolaou (1988) and Burridge and Chang (1989) to allow for tunneling P waves for S-wave incidence beyond the critical angle. When the reflection coefficients at the interfaces are scaled as 1/√ N while N → ∞, and when time is measured in units of vertical travel time across an average layer, numerical solutions of the exact problem show that the shape of the broad transmitted pulse approaches the limiting form given as the solution of a certain integrodifferential equation in accordance with our asymptotic theory.