Transport theory and boundary-value solutions I The Bethe–Salpeter equation and scattering matrices

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Abstract
The mutual coherence function of a wave in a random medium with one rough boundary is considered, and the governing equation is obtained in a form of the Bethe-Salpeter (B-S) equation, such that the medium and the boundary are involved on exactly the same footing. The method is based on a previous paper for a purely random boundary, and the present B-S equation is a generalized version of the governing equation when the medium is also random. The equation is compared with the conventional method of solving the transport equation, showing the correspondence of each quantity in detail, and it is concluded that they are equivalent to a sufficient accuracy in the case of one boundary. To obtain the solution, incoherent scattering matrices are introduced independently for both the boundary and the medium, where the former has a direct connection with the conventional scattering cross section per unit area and the latter may be obtained by solving the transport equation. Hence the solution is obtained in several expressions in terms of an effective scattering matrix of the boundary as affected by the random medium and also in terms of a corresponding matrix of the medium free from the boundary. The previous expression by Fung and Eom [IEEE Trans. Antennas Propag. AP-29, 899 (1981)] is reproduced to practical accuracy by the present exact formulation.