224 Lecture 12 
the sine in the last expression may amount to many multiples of 7. The phase 
of the contributions from the various patches of inhomogeneity may, therefore, 
be considered to be random too, and the sine may be replaced by its rms value, 
1/J2. The result, then, reduces to 
V, =ak VRE (57) 
Practically the same formula has been derived by Mintzer [30] on the basis of 
a Gaussian correlation function. The range of validity of this solution is usually 
called the "interference" or "wave-theory" range of forward scattering. 
As the frequency increases, the scattered radiation becomes more and more 
concentrated in the forward direction, and the phase differences between the 
extreme contribution left and right of the axis of the main beam become neg- 
ligibly small. Essentially, then, forward scattering is a focusing or defocusing 
effect. Within this range, L/kR? <1; and the sine above can be replaced by its 
argument 
¥2 
a <a?>” IRL sat - <as(E) (58) 
Except for an insignificant factor, the last expressionis identical with the Berg- 
mann formula for ray-theory limit of scattering [16]. Within the ray-theory 
range, the fluctuations of the signal are mainly caused by focusing and defocusing 
effects. They increase with the % power of the range and are independent of the 
frequency. For large values of the range (see Fig. 12.15), the solution passes 
over into the wave-theory result. The phase differences between the various 
scattered rays accumulate, and the focusing and defocusing are destroyed by 
the random phases of the scattered pressure contributions; the signal fluctua- 
tions are due entirely to interferences. The transition between ray- and wave- 
theory range occurs when 
= Eg SkRo (59) 
If this condition is fulfilled, the diffraction cones generated at the boundaries 
of the scattering patches cover the entire cross section of the beam. The 
receiver becomes surrounded by an interference region, and the scattering 
patches as seen from the receiver have radii approximately equal to the central 
zone in the Huygens zone construction; patches of this diameter are particularly 
effective in generating signal fluctuation. 
12.4.2. Computation of Amplitude and Phase of Fluctuations for Spherical Patches 
The fluctuations of the amplitude and the phase of the transmitted signal 
depend not only upon the magnitude but also upon the phase of the scattered 
pressure. The computations in Section 12.3 seem to leaa to the result that the 
scattered pressure is always in phase with the incident sound. This conclusion, 
however, is correct only for distances from the scatterer that are very large 
compared to the diameter of the scatterer, so that the square and the higher- 
order terms in the exponent ofthe Rayleigh integral can be neglected. At smaller 
distances the scattered pressure may have to be 90° out of phase, as has been 
demonstrated for the infinitely large scattering slab [Eq. (49)] or any phase 
between zero and a multiple of 27, as has been found for the finite scattering 
