Wave propagation in media with time‐dependent spatial inhomogeneities

Abstract

The propagation of waves in a medium with time‐dependent spatial inhomogeneities in its velocity profile is given a rigorous integral equation formulation for source and incident field descriptions. This is in the spirit of Rayleigh, whose aim was to bypass complications arising from boundary conditions. A perturbation expansion, correct to all orders, is then obtained with the aid of a simple identity, for the two cases: (i) weak scatterers and (ii) strong scatterers. It is shown that for case (i) a correction term has to be added to previous results already in second order—this seems to have been overlooked until now. For case (ii), the expansion obtained here is new—the reason for this gap in the existing literature is discussed. With these results, expressions are derived for the wavefunctions when the inhomogeneity in the medium is due to distributions of moving particulates. These new closed‐form time‐dependent approximations exhibit explicitly the appropriate Doppler behavior. The application of this formulation to other cases, such as media with random inhomogeneities, is discussed.