Excellent theoretical works have been accomplished over the past two decades on the scattering of acoustic waves by randomly rough surfaces. In most of those works the starting point has been the approximation of surface boundary conditions by the physical-optical approximations, except in the cases of Wagner and Lynch. The former analyzed the shadowing phenomena of randomly rough surfaces and the latter considered improving the surface field expression based on the physical-optical approximation by making curvature corrections. A basic assumption for irregular surfaces which is common to all the work is that the roughness, i. e. , deviation from the mean level is given by a single Gaussian distribution. On the other hand Beckmann investigated the scattering of electromagnetic waves by a composite rough surface whose roughness is expressed as the sum of several independent Gaussian random functions. This paper is an extension to the case of the scattering of acoustic waves by a composite rough surface whose height function is given as the sum of a finite number of statistically dependent Gaussian random variables [Eq. (31)]. The sea-bottom is made up of the basic profile (continental shelf) and the small-structure roughnesses caused primarily by rocks, coral reef, pebbles, sands, mud, sea weeds, etc. , and the natural evidence about the sea-bottom indicates that the small-structure roughnesses are definitely dependent upon the basic profile, i. e. , the large-structure roughness. In view of this the statistical model of rough surfaces proposed here is thought of as more natural and better than the ones presented by Beckmann and others. A coordinate system for describing the scattered field is shown in Fig. 1, and Fig. 2 explains the geometry of the scattering. It is assumed that the wave equation (7) holds throughout the medium bounded by the rough surface and the sphere at infinity, and Green's theorem provides Eq. (8) for the reflected pressure φ_r(P) at an observation point P. For simplicity the boundary condition at the surface is taken to be zero pressure - the "free surface" condition. Following Beckmann's approach the mean intensity of the scattered pressure wave is obtained as Eq. (32), where ζ is the height function given by Eq. (31). ζ_p(p=1, 2, . . . . , n) are Gaussian random variables with zero means and variances σ_p^2. It is also assumed that ζ_p is correlated with ζ_q by the correlation function ρ_<pq>. The characteristic function in the integrand of Eq. (32) can be expanded as Eq. (36)(see Rice), where C_<pp>(τ), C_<pq>(τ) are correlation coefficients given in Eqs. (38). The particular forms of the correlation coefficients are known to fit quite well many contour maps. For a very rough surface for which the relation (39) holds the mean intensity of the scattered pressure wave is now given as in Eq. (42). For a rough surface which may be considered to consist of a large-structure roughness and a small-structure roughness n=2, and Eq. (42) yields Eq. (44).
