III-84 



Then using the properties of the delta function, we obtain 



p (x, y, z) = 7 A e 

 ^sc ' / J n 



.k(a^x+Y^z) (III-94) 



n=-aD 



with OL and y given by (III-62). Thus, we obtain an expression giving Pg^, as 

 a sum of plane waves having the direction cosines obtained for general reflection 

 from a periodic surface . In particular, suppose the surface is given by 



S(x) = hcospx. Since 6^^^^°^^^ is the generator for the Bessel function, cf(III-71), we 

 can write (III-92) as a series in Bessel functions. Substituting this in (III-93), 

 and after some analysis, we obtain the coefficients 



(a - a )^ + (y - Y )^ f- -, 



A = S 2_ J (Y-Y)kh (III-95) 



n o / \s n L n J 



2y^(y-Y^) 



The basic limitation of this approach is associated with the shape of the uneven 

 surface. The nature of the irregularities must be such that overshadowing does 

 not occur, i.e., the surface should be locally flat. Overshadowing is assumed 

 to occur when the angle of incidence is greater than the angle between the normal 

 to the plane z=0 and the tangent to the surface at the point of maximum slope. 

 The larger the slope, the smaller must be the angle of incidence to prevent over- 

 shadowing. The criterion of local flatness, when applied to those portions of the 

 surface with the least radius of curvature, can be shown to reduce to the in- 

 equality 



2 



cos a>> i ^ (III-96) 



where a is the angle of incidence . 



We first discussed the Rayleigh formulation which was based on the 

 assumption that kh < < 1, where k is the wave number of the incident sound and 

 h is the height of the surface roughness . The Brekhovskikh model just con- 

 sidered assumes that in a neighborhood of the surface the reflected sound follows 

 the laws of geometric acoustics and behaves as though scattered from a plane 

 tangent to the surface. Another formulation, due to Lysanov (Ref . III-22), is 

 based on the assumption that the scattering surface satisfies the conditions 



2 ^dx^] 



kh ^ < <1 (III-97) 



ox 



artbur B.llittlcllnt. 



S-7001-0307 



