Several expressions for the mutual coherence function are obtained for waves scattered by a random layer with rough boundaries (including those waves transmitted through) by modifying the previous Bethe-Salpeter (B-S) equation for a random medium with one rough boundary. Considering that the medium and the boundaries are involved in the equation on exactly the same footing, two sets of addition formulas are prepared for the incoherent and coherent terms, respectively, to obtain various expressions of the mutual coherence function by exchanging the roles of the medium and the boundaries. The conventional method of solving the transport equation is shown to be available only when the separation of the boundaries is sufficiently large compared with the coherence distance of the wave, whereas the expressions obtained are available beyond this limit. The previous expression by Fung and Eom [ IEEE Trans. Antennas Propag. AP-29, 899 ( 1981)] is also shown to be reproduced with the present exact formulation, as long as one of the boundaries is perfectly random. The corresponding expressions for fixed scatterers embedded in a (homogeneous or semi-infinite) random medium are also obtained, which show an explicit shadowing effect. Summarized in the appendixes are basic equations including addition formulas, the Green function and the scattering matrix of the B-S equation, and a general theory of fixed scatterers embedded in a random medium.
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