On modeling discontinuous media. Three-dimensional scattering

Abstract

A realistic model for the propagation of scalar waves in a medium must take into account discontinuities of the medium parameters, which may act as ‘‘hard’’ reflectors; discontinuities of their gradient, which may act as ‘‘soft’’ reflectors; and continuous variations of parameters. The direct scattering problem is presented here. The ‘‘mixed potential–impedance’’ equation that characterizes the model depends on two arbitrary parameters: one of them would be the potential if the equation reduces to the Schrödinger equation; the other one would be related to the impedance if the equation describes more ‘‘classical’’ waves. After the mathematical tools are constructed (Green’s functions, etc.), a rigorous three-dimensional scattering theory is described. It encompasses the quantum three-dimensional scattering theory and the theory of acoustical scattering (for instance) by systems of regular surfaces of arbitrary shape. The main integral equations of the quantum scattering theory are generalized. Scattering amplitudes due to reflectors and scattering amplitudes due to diffuse scattering after reflectors have been taken into account are defined and constructed. Born and quadratic approximations are discussed: the explicit formulas corresponding to the scattering by discontinuities and those corresponding to diffuse scattering are not reducible to each other exactly (i.e., unless filtering and errors are allowed). The results can also be used to described rigorously the three-dimensional scattering by the ‘‘wave-equation’’ in the frequency domain—and in particular the response to an impulsive localized source. Further generalizations are in progress.