Ill- 82 



=1^ - ia-wpr -h" r (III-85) 



appear, where a = k sin 9 cos 6 and b = k sin sin 4, and (r, 9, 6) are a system 

 of spherical coordinates . No computations or experimental results are given . 

 Using this model, Heaps also obtains expressions for the intensity and average 

 intensities of the resultant field. 



The so-called Rayleigh approximation requires that the surface irregu- 

 larities are small in comparison with the wavelength of the incident radiation. 

 Brekhovskikh (Ref . III-18) has developed a theory which permits large irregulari- 

 ties, but assumes that at the irregular surface the sound field can be specified in 

 terms of the laws of geometric acoustics . At each point on the surface, the sound 

 field is assumed to be the same as if the reflection from that point were to occur 

 from an infinite plane tangent to the surface at the specified point. In other words, 

 the distribution of the scattered field over the surface is assumed to be that which 

 could be expected on the basis of considering only one incident wave while neglect- 

 ing secondary scattering from individual sectors of the surface. By specifying the 

 field at the surface in this manner, it is possible to use the Green (or Helmholtz) 

 formula to compute the field at points away from the surface . 



Consider a surface of the form z = S(x) and an incident wave given by 



ik(ax+Yz) (III- 86) 



p . (x, z) = e 

 ^mc 



I 



At the surface the scattered wave p is given by the boundary condition: 



p^Jx,S(x)] + p^Jx,S(x)] = 0, 

 Thus, we have 



p^^[x,S(x)]^-e^^t°^^-^YS^^)] (III-87) 



9p 

 sc 

 We also require -^ , where the derivative is taken along the normal to the sur- 

 face. Using the condition of locally specular reflection, this derivative is given 

 on the surface by 



^Psc .,, , ik[ax+YS(x)] (III-88) 



-r — = ik(aa+ Yc) e '■ 

 o n 



where (a, c) are the direction cosines of the normal to the surface. Let £1 = 

 Xi, yi, S(xi) be a point on the surface, r = (x, y, z) be an arbitrary observation 

 point, and k = (k^, k , k ) be an arbitrary propagation vector. We use the 

 relation 



^artbur m.lLittlcJnt. 



S-7001-0307 



