IV-13 



We now introduce t±ie approximation (IV-43) into the scattering integral (IV-39) and 

 obtain for the far field the expression: 



ikx 

 4Trx 



Pi M--^S^ j d? e"^^^^ u(?) (IV-45) 



V 

 Equation (IV-45) leads to the following observations: 



1. The scattered pressure field has the nature of a spherically spreading 

 wave with a multiplicative directivity pattern which depends on the angle 9 . In 

 particular, the forward direction of scattering the vector d is zero, and the pressure 

 is given by: 





2k=e^^M -- ,.. Se 



V 



In this equation, we have introduced the parameter S as the forward scattering 

 strength of the inhomogeneity. We note that the forward scattering strength is pro- 

 portional to the square of the frequency and to the total volume integral of the 

 correction to the index of refraction. The pressure distribution in the far field in 

 any direction may now be written in terms of a directivity pattern as: 



ikx 

 pi (x) = SD (9, (4) (IV-46b) 



Here we have introduced D(0, d), the directivity pattern of the scattering, * which 

 corresponds to the ratio of the integrals in (IV-45) and (IV- 46b). 



2. We recognize the integral in (IV-45) to be the Fourier transform of the 

 refractive index u (£) evaluated at a wave number vector kd which has a magnitude 

 2k sin y . The Fourier transform of u (^), e.g. , N (x), represents the amplitude of 

 the decomposition of u (?) into waves with wave-number vector x; in other words, it 

 is the spectrum of j (^). If the spectrum N (x) is large in vicinity of x, it corresponds 

 to an inhomogeneity with a substantial Fourier component of wave-number X- Equation 

 IV-45 tells us that the scattered field will be large for a scattering angle 9 such that 

 the spectrum N (x) is large at x = 2k sin £. . If the spectrum of u (£) is large only for 

 values of x much less than k, it is clear that the substantial portion of the scattered 

 field will be confined to a small angle 9 ~^ • In other words, inhomogeneities which 



* 9 and (i are the polar and azimuthal angle of a spherical coordinate system, 

 x = (x, 9, i)) with xi as polar axis. 



S-7001-0307 



