IV -41 



It is clear from the above that the expected value of the phase will be just the phase 

 of the incident wave Sq. The variance of the phase is therefore given by <Si^>, 

 and this is one of the averages which we must calculate. 



We are also interested in the variance of the amplitude of the total pres- 

 sure. The literature is full of erroneous derivations* of the variance, and we shall 

 therefore take some time to develop an expression which has somewhat wider 

 applicability. 



Since (IV-36) shows the fluctuation of the complex phase to be the sum of 

 a large number of independent contributions (independent if we divide the volume V 

 into a number of sub-volumes each of which are still large compared to a single 

 patch), we would expect, according to the central limit theorem, both the real and 

 the imaginary part of K to obey a normal distribution. In fact, we have pretty 

 good evidence (see Figure IV-7) that u itself is already normally distributed, which 

 would even make the invocation of the central limit theorem unnecessary. Si and 

 Bi therefore obey a multivariate normal distribution with a characteristic function 

 which is: 



/<;Si^> <SiBi>\ /o 



X 



(a, 6,.<e"'Si+»a>=e \<SiB>><B,^ >, ,. , „y_,3^ 



The computation of any averages involving the total pressure or its amplitude is 

 much facilitated by the use of this characteristic function. We observe that the 

 amplitude of the pressure is given by 



A(x) = |p(x)| - A (x)e^^^-^ (IV- 84) 



— — o — 



Averaging this expression over the ensemble, we obtain: 



<A> = A <e^^^^^> = A Y(o, -i)^A e^^^^'^ (IV-85) 



o o /\ o 



"The common practice is to regard the total amplitude as very little different 

 from the amplitude of the incident wave, expand the total amplitude to first order 

 in the small quantity, and then square it to obtain the square of the amplitude. 

 Taking the average, the first order term drops out and there remains the second 

 order term. This second order term is incorrect, since it does not include the 

 product of the zeroth and second order terms of the original expression of the 

 total amplitude. 



Arthur BXittk.lint. 



S-7001-0307 



