Validity of a selective summation technique in analysis of wave propagation through random media

The so-called moment equation for a successively forward scattered wave was derived under a definite condition. The condition shows that the applicability of the equation is much more extensive. When a selective summation technique is applied to the derivation of the moment equation, another condition besides the above is necessary for the existence of the equation. The extra condition is due to the selective summation of scattered waves and shows the validity of the technique for the nth moment of forward scattered waves. For example, if the waves propagate along the z axis, a necessary condition for the technique to be valid for the first, second, and fourth moments is k²l²B(0, 0) ≪ 1 where k is the wave number in free space and l and B(r, z) are the correlation length and the correlation function of fluctuation of the medium, respectively. For all the higher moments the technique is applicable only to the case of B(r, z) = B_r,(r)δ(z) where δ(z) is the Dirac delta function.