OBSERVED FLUCTUATION 



159 



deviations luis the same dimension as an amplitude. 

 It is called the root-mean-square (rms) deviation of 

 the aniplitucU^ or, more briefly, the standard devia- 

 tion of the amplitude. If it is divided by the mean 

 amplitude, the rt;sulting dimensionless quantity is 

 called the relative standard deviation of the ampli- 

 tude; this ((uantity is often expressed in per cent. 



The analogous quantities formed with intensities 

 and levels bear analogous names. These names are, 

 in fact, common in all fields of statistics. If the rela- 

 tive standard deviation of the amplitude is very 

 small compared with unity, the relative standard 

 deviation of the intensity is about twice the relative 

 standard deviation of the amplitude, while the abso- 

 lute standard deviation of the level is approximately 

 4.34 times the relative standard deviation of the in- 

 tensity." The fluctuation of underwater sound is 

 usually so large that these relationships between the 

 standard deviations do not hold. 



Relative standard deviations of the amplitude 

 have been determined for transmitted signals of 

 underwater sound under various conditions.'"' Most 

 of the available data were taken at 24 kc. From the 

 data at that frequency, an analysis was made of the 

 dependence of the relative standard deviation on 

 refraction conditions.^ It was found that in the 

 presence of strong downward refraction the median 

 of the relative standard deviation, for the 29 samples 

 collected, was 38 per cent.'' For eleven samples, in 

 which the receiving hydrophone as well as the pro- 

 jector were located within a mixed layer above a 

 thermocline, the median of the relative standard 

 deviation was 47 per cent. Seventeen samples, in 

 which the hydrophone was in the thermocline be- 

 neath a mixed layer, showed a median relative stand- 

 ard deviation of 41 per cent, not much higher than 

 the fluctuation in the presence of strong gradients 

 from the surface down. Although these differences are 

 probably significant, they shoxdd not be overesti- 

 mated, in view of the wide spread within each of the 



" 4.3429 is 10 log e where the log is to the base 10 and e is 

 the base of the natural logarithms. 



'' In the discussion of the spread of a given set of data, it is 

 very convenient to use the terms "median" and "quartile." 

 Their meaning is as follows. If all the determinations of a cer- 

 tain quantity are arranged in the order of increasing magni- 

 tude, the value corresponding to the midpoint of the array is 

 called the median value of the spread. The point separating 

 the lowest quarter of determinations from the rest is called 

 the lower quartile, and the point which separates the highest 

 quarter of all determinations from the rest is called the upper 

 quartile. These terms will be used occasionally in the re- 

 mainder of the chapter. 



groups of samples discu.s.sed previously. The lower 

 and upper quartiles in the group of strong downward 

 refractions are 4(j per cent and 30 per cent, respec- 

 tively, while the ([uartilesforthe isothermal group are 

 61 per cent and 44 per cent. It is probably justifiable 

 to .say that, on the average, the amplitude fluctuation 

 in isothermal water is significantly higher than the 

 amplitude fluctuation in the presence of strong down- 

 ward refraction. The width of the quartile spread 

 shows that even under .similar conditions the magni- 

 tude of the fluctuation itself fluctuates from sample 

 to .sample. In view of the large number of signals 

 making up a sample, usually between 50 and 200, this 

 variability is not to be explained as sampling error 

 but represents an actual change in the transmission 

 conditions as they affect signal fluctuation. The high 

 degree of variability of fluctuation is an indication 

 of the complexity of the underlying mechanism or 

 mechanisms as well. 



Some information is available concerning the de- 

 pendence of the relative standard deviation of the 

 amplitude on frequency. One set of experiments, 

 carried out at UCDWR, involved the simultaneous 

 transmission of signals at two supersonic frequencies.^ 

 The frequency pairs used were 14 and 24 kc, 16 and 

 24 kc, 24 and 56 kc, and 24 and 60 kc. In 17 runs one 

 frequency was either 14 or 16 kc, while the other was 

 24 kc. It was found that the mean of the relative 

 standard deviations at the lower frequency (14 or 

 16 kc) was 38.8 per cent and at 24 kc 37.7 per cent. 

 The difference is well within the root-mean-square 

 spread, and is thus not significant. For the individual 

 samples themselves, the difference between the fluc- 

 tuations at the two frequencies is considerable for 

 some runs, amounting to 19.2 per cent in one case. 

 The root mean square difference is 8.5 per cent. As a 

 result, it may be concluded that the average fluctua- 

 tion is the same at 15 and at 24 kc, but that for any 

 individual run the fluctuation may be considerably 

 different at these two frequencies. In the majority of 

 cases, however, high fluctuation at one frequency is 

 associated with high fluctuation at the other, and 

 unusually small fluctuation at one frequency tends 

 to be associated with small fluctuation at the other. 

 An analysis of the runs carried out with the frequency 

 pairs 24 and 56 kc, and 24 and 60 kc, leads to similar 

 conclusions for these frequencies." 



" The correlation coefficient between the magnitude of the 

 fluctuation for the frequency pair 14, 16 to 24 kc was found 

 to be0.65andforthefrequencypair24and56 or60kc,0.68. For 

 a definition of the coefficient of correlation, see Section 7.2.3. 



