160 



INTENSITY FLUCTUATIONS 



At frequencies below 10 kc, it was found both at 

 San Diego and at the New London laboratory of 

 CUDWR that the magnitude of the fluctuation de- 

 creases with frequency. No quantitative data are 

 available. A particularly interesting result was ob- 

 tained in a single transmission run at 5 kc at San 

 Diego. It was found that at moderately short range 

 the relative standard deviation of the amplitude was 

 47 per cent, while beyond the computed last maxi- 

 mum of the image interference pattern (see Chapter 

 5) it dropped to 10 per cent. 



1.25 



1.00 



M X 



2 "^O-TS 



z t 



2 5^0.50 



< < 



iO.25 



0,5 



1.0 



SIGNAL AMPLITUDE IN ARBITRARY UNITS 



Figure 1. Cumulative distribution function of four 

 signals. 



Complete evidence is not available concerning the 

 dependence of the magnitude of fluctuation on range. 

 It is known that at distances of a few feet fluctuation 

 of transmitted sound is negligible. From 100 yd out 

 to very long ranges the average magnitude of the 

 fluctuation appears to be the same at all ranges. No 

 analyses have been made comparing the magnitude 

 of fluctuation at different ranges under identical 

 thermal conditions. 



There have been recent experiments at UCDWR 

 designed to determine the possible dependence of 

 fluctuation magnitudes on the depths of the projector 

 and receiver. In these experiments, a cable transducer 

 was used as a projector which could be lowered to 

 various depths up to 300 ft. When the projector depth 

 was kept constant at 16 ft, the magnitude of the 



fluctuation was found to be independent of hydro- 

 phone depth (except for MIKE patterns). ^ When 

 both the projector and receiver are desp, it is possible 

 to distinguish between the direct and the surface-re- 

 flected signal. Two runs were carried out with the 

 projector at a depth of 140 ft and the hydrophone at 

 a depth of 300 ft, and the direct and surface-reflected 

 signals were analyzed separately. For the direct 

 signal, the relative standard deviation of the ampli- 

 tude was 9.8 per cent for the first run and 6.8 per cent 

 for the second run; while for the surface-reflected 

 signal the two fluctuations were 57 per cent and 51 

 per cent respectively. With both projector and hydro- 

 phone at a depth of 300 ft, the fluctuation of the 

 direct signal amplitude was 6.0 per cent and of the 

 surface-reflected signal 50.5 per cent. These results 

 indicate that much if not most of the observed fluc- 

 tuation is caused by mechanisms operating at or near 

 the sea surface. The remaining fluctuation is proba- 

 bly caused at least in part by the slight directivities 

 of the cable-mounted projector and receiver used in 

 these experiments. 



7.1.2 Probability Distributions 



The probability distribution of a set is that func- 

 tion which tells how many members of the set lie 

 between two specified values. Suppose, for instance, 

 that we consider a sample of signals transmitted 

 consecutively over the same transmission path . After 

 these samples have been rearranged in order of in- 

 creasing amplitude, it is then easy, by mere counting, 

 to say how many of those signals have amplitudes 

 less than ai, how many have amplitudes less than a2, 

 and so on. If we divide these numbers by the total 

 number of members of the sample, we obtain the 

 fraction of signals with amplitudes less than a as a 

 function of a, say P{a). P{a) vanishes for a = 0, 

 equals unity for a = oo, and increases steadily be- 

 tween these limits. This function is called the cumu- 

 lative or integrated distribution function. As a very 

 simple case, Figure 1 shows the integrated distribu- 

 tion of four signals with amplitudes of 0.2, 0.4, 0.5, 

 and 0.7. In the theory of statistics, it is usually as- 

 sumed that if the number of members of the set is 

 increased without limit, the shape of the function 

 P{a) approaches a limiting shape in better and better 

 approximation. It is this limiting shape to which a 

 fundamental physical significance is ascribed. If we 

 assume that the limiting function can be differenti- 

 ated, then 



