OHSERVED FLUCTUATION 



163 



lations, then llic di.stribution function can be pre- 

 dicted except for one single parameter, and this 

 parameter is the average intensity. The Rayleigh 

 distribution differs in this respect from a Gaussian 

 distribution, which contains two adjustable param- 

 eters, the average value and the standard deviation. 

 The other distribution function to be discussed 

 here is the image interference distribution. It is calcu- 

 lated on the assumption that all the fluctuation of 

 transmitted signal intensity is caused by the random 

 interference of the sound transmitted directly and the 

 sound reflected from the surface of the sea. The ex- 

 tent to which this assumption is justified will be dis- 

 cussed in Section 7.2.2. Let us assume that the ampli- 

 tude of the direct signal alone is ai, while the ampli- 

 tude of the surface-reflected signal by itself is 02. If 

 the phase angle between these two components is de- 

 noted by d, the resultant amplitude a will be given 

 by the expression 



a^ = al -{- al -\- 2aia2 cos d. (17) 



With the large-scale geometry given, the values of ai 

 and Oi will not vary significantly; but the phase dif- 

 ference between the two paths, 6, will change ran- 

 domly because of the action of waves and because of 

 the minute changes in position of both vessels. In 

 other words, while ai and 02 will be treated as fixed 

 parameters (the values of which can, however, be 

 specified only if the depths of source and receiver and 

 their distance are known), 6 will be assumed to take 

 all values between — tt and tt with equal probability. 

 Since the value of a does not depend on the sign of d, 

 we shall restrict ourselves to values of d between 

 and -\-K. The probability that exceeds a certain 

 value d* equals 1 — d*/Tr, that is, the cumulative 

 distribution function for A satisfies the equation 



P{a) 



1 - 



(18) 



where a and 6 are related to each other by means of 

 equation (17). In other words, the fraction of signals 

 for which the phase angle exceeds the value 6 is 

 identical with the fraction of signals with an ampli- 

 tude less than a. By differentiating both sides of equa- 

 tion (18) with respect to a, we obtain an equation 

 for the differential distribution, p(a), 



P(a) = --^' (19) 



da 



with 



de 



^«~ V-(a? - alf + 2{a\ + al)a? - 



-2a 



(20) 



from equation (17). We find, then, for p(a) the ex- 

 pression 



a 



2 



via) = - — — 



air- + 2(a? -I- al)a' - a* 

 1^1 — o-i] ^ a ^ en + ctj. (21) 



Outside the limits indicated, p(a) vani.shes since the 

 amplitude cannot be greater than oi -|- a^ nor less 

 than |oi — ttil. For P{a) we find, by means of the 

 relationship 



^^«) = L 



the expression 



Pia) = 



a = \ai—a2\ 



11 . a2 

 — \- - arc sm — 

 2 X 



p{a)da, 



(22) 



(a? + at) 



2a 102 



= - arc sin 



|/°- 



- (ai — ch)^ 



4aia2 

 ai — ail ^ a ^ tti -|- 02 



(23) 



by trigonometric transformations. Both expressions 

 (21) and (23) become much simpler if it is assumed 

 that the reflection from the sea surface is perfect, that 

 is, if ai = a2. We have, then 



via) 



1 



'rVial 



^ a ^ 2ai, (24) 



and 



P{a) = - arc sin ;^ ^ a g 2ai. (25) 

 T 2ai 



At UCDWR, some experimentally obtained cumu- 

 lative distribution functions of transmitted signals 

 have very nearly the form of equation (25), while 

 others are approximated by a Rayleigh distribution. 

 All the distribution functions published at UCDWR 

 are plotted as integrated distributions. This has been 

 done because with a limited size of the sample the 

 differential distributions would be very difficult to 

 compute with any degree of reliability. Integrated 

 distributions are reasonably accurate in tlie central 

 part of the curve, but the "tails" at both ends are 

 necessarily based on very few experimental data. 

 This is unfortunate, because the gross features of 

 integrated distributions, and particularly the central 

 portions, are not very sensitive to changes in the 

 character of the distribution. By definition, all inte- 

 grated distributions are functions which increase 

 steadily from zero at — 0° to 1 at -|- 00 . The central 

 portions of two different distribution functions will be 

 determined in their gross appearance by the location 



