A Comparison of Spectral-Domain Methods with Direct Solutions to the One-Dimensional Wave Equation

Abstract

This paper compares the first- and second-order moments computed by applying two distinctly different methods to the one-dimensional wave equation. The transmission-line form of the one-dimensional wave equation is a special case of a spectral-domain formulation that we developed earlier, but the principal results derived in this paper can be generalized to three dimensions. An essentially complete statistical description of the scattered wave field can be obtained from the method of invariant imbedding, but that method is difficult apply to the three-dimensional problem. The more restrictive Markov approximation, which is used to characterize forward scattering in continuous random media, can be applied in general treatments that accommodate backscatter. We show first that the closure hypothesis obtained by using the Novikov-Furutsu theorem provides a closed system for computing field moments that is correct to terms that are second-order in the perturbation to the constitutive parameter. The Markov approximation is then used to simplify these equations, whereby simple solutions are obtained. We compare these results to the more general results obtained from invariant imbedding. The effective wavenumbers for the coherent wave fields obtained by the two methods differ only by a phase factor that can be neglected under the Markov approximation. For the second-order moments, we develop some general constraints that apply to all solutions, and then show that the Markov approximation leads to the equations of radiative transfer ; however, the radiative transfer equations agree with the more general solution only if an optical-depth parameter is less than unity. The ramifications of these results are discussed.