E. J. Skudrzyk 221 
and the integral approaches a finite limit. It is possible to derive approximate 
expressions for this limiting value. However, there is not much point in doing 
so, since the computation would be equivalent to a computation of the phase 
change of the signal because of its different velocity in the medium. 
In the Kolmogorov case, m= fy and the resulting solution is represented in 
the last line of Table 12.11; the Kolmogorov equivalent for the patch radius R is 
the depth h of the measurement, since the depth is the only dimensional param - 
eter that is available and x) =7/2h. In the curve for the Kolmogorov case in 
Fig. 12.13, therefore, R has to be interpreted as 4A. 
Figure 12.13 compares the scattered intensities for the case of practical in- 
terest. The most frequently usedcorrelation functionis the exponential e~/* since 
integrals based on this function can usually be evaluated. Also, the exponential 
correlation function seems to lead to good agreement with the experimental re- 
sults in a great number of practical cases. Figure 12.13 shows that the expo- 
nential function generates about the same amount of scattering as the function 
R(p)=1-(p/R)?, r<R. Unfortunately, the exponential correlation function leads 
to divergent integrals for the ray-theory limit of scattering. The curve for the 
Kolmogorov case is slightly broader; therefore, the corresponding correlation 
function is slightly steeper. This slight increase in steepness in the Kolmogorov 
case seems to be sufficient to ensure convergence of all the scattering integrals. 
Frequently, the Gaussian correlation function is used as a substitute for the 
exponential one to enforce convergence of the scattering integrals, Figure 12.13 
shows that the Gaussian function represents a poor approximation to cases where 
an exponential or a Kolmogorov correlation function would be expected to apply. 
12.4. FLUCTUATION OF THE TRANSMITTED SIGNAL 
12.4.1. Elementary Theory 
The amplitude of the transmitted signal fluctuates continuously because of 
the focusing and defocusing of the incident sound by the continuously moving 
patches of inhomogeneity and because of interferences caused by the scattered 
pressure. These signal fluctuations can be computed with the aid of the Rayleigh 
integral. 
The transmitted sound is given by the vector sum of the directly transmitted 
pressure and the scattered pressure. Therefore, the phase of the scattered 
pressure must be taken into account. Forvery large scatterers and at great dis- 
tances from the scatterers, the scattered pressure is found to be displaced in 
phase by 90° with respect to the transmitted sound and affects only the sound 
velocity in the medium. The change in amplitude is then a second-order effect. 
For large scattering patches, it can be shown that the second-order terms have 
to be retained in the exponent of the scattering integral [Eq. (20)] to obtain the 
correct value of the scattered pressure phase. This fact complicates the com- 
putation of the transmitted signal considerably. 
Up to the present, all the computations that have been made have been based 
on a Gaussian correlation function, or on the assumption of a definite patch size, 
both of which are very unrealistic assumptions. Since the mathematical theories 
of the transmitted signal fluctuation are so extremely complicated, they cannot 
