IV -37 



(IV-77) shows that the larger patches of the inhomogeneities are important in 

 determining the forward scattering properties, whereas patches of the order of 

 half the wavelength of the incoming sound dominate in determining the back 

 scattering. 



The total power scattered by a unit volume of scattering inhomogeneities 

 may be obtained by integrating (IV-77) over the surface of a semisphere: 



^2 



P = 2n 



forward 



de x^<| pi (xiB)! ^> sine=nk^ 



/2 E (2k sin 9/2) 



"^ ^'"^ (21. sin 9/2)" 0^-'« 



Because of the trigonometric identity cos 9 = 1-2 sin 9/2, we can write the 

 differential sin 9 d 9 as 2d sin^ 9/2. We therefore introduce the new variable 

 y = (2k sin 9/2)^ in (IV-78) and obtain a very simple expression for the total power 

 scattered forward by unit volume of scatterer: 



2k^ 



f E (^/^) 



P, . = ? k^ dy ^ (IV-79) 



forward 2 J ^ y 



o 



Consider now a plane wave incident on an infinitely extended slab of scatterers, the 



slab having unit thickness. A moment's reflection will make clear that (IV-79) must 



also be the total scattered power received per unit of surface at any point behind the 



slab. If the slab has a thickness L, the total scattered power will be: 



2k^ 



E (Vy) 



P = ^k^L I dy-^:^ (IV-80) 



l-^\ 



since we may simply add the scattered power from each of the individual slabs of 

 unit thickness. Thus, an observer, embedded at a range L from the surface of 

 an inhomogeneous half- space will find that (IV-80) determines the scattered power 

 which he receives from forward scattering of the intervening medium if a plane sound 

 wave is normally incident on the inhomogeneous half- space. With the use of 

 Table IV-2, we may work out the details of (IV-80) for the case of an exponential 

 and a Gaussian correlation function. These are presented in (IV-81). 



artbur 2l.littlcJnt. 



S-7001-0307 



