The formation of caustics in two- and three-dimensional media

Abstract

In terms of ray theory, the focus point (also related to caustics and triplications) is the point in space where the ray position is stationary for perturbations in the initial condition. Criteria for the formation of caustics are presented. With ray perturbation theory, a condition for the development of triplications is defined for plane wave sources and for point sources. This theory is then applied to two cases of slowness media: 1-D slowness perturbation models and 2-D Gaussian random media. The focus position in 1-D slowness models is proportional to the inverse of the square root of relative slowness fluctuations. For Gaussian random media, the distance at which caustics generate is dependent on the relative slowness perturbation to the power of −2/3. We use snapshots of propagating plane wavefields to show that caustics develop as predicted by theory. The theory for caustic formation can be generalized to three dimensions.