CAUSES OF FLUCTUATION 



171 



loads to the result tluit if the refracting properties of 

 tlie microstructure were alone responsible for fluctua- 

 tion, the magnitude of fluctuation should increase 

 with range. At moderate ranges the magnitude of the 

 fluctuation should be proportional to the 1.5th power 

 of the range. Since this hypothesis is based on ray 

 acoustics, the fluctuation should be independent of 

 the frequency, as long as the wavelength is short 

 enough for ray acoustics to be applicable. 



The dependence of fluctuation on range predicted 

 by this hypothesis has not been confirmed by obser- 

 vation.s, although the variability of the magnitude 

 of the fluctuation is so great that a small effect might 

 not have been discovered. For that reason, some de- 

 pendence of fluctuation on range cannot be definitely 

 ruled out. The theoretical formula connecting the 

 magnitude of the predicted fluctuation with the 

 parameters of the microstructure appears to lead to 

 a fluctuation of a magnitude much smaller than ob- 

 served. There is, however, one feature which appears 

 to suggest that refraction by microstructure is at 

 least a contributing cause of the observed fluctuation. 

 It was pointed out that fluctuation caused by micro- 

 structure should be frequency-independent for a wide 

 range of frequencies. In this respect it differs from 

 hypotheses based on interference, since interference 

 leads to fluctuation which is critically dependent on 

 frequency. It has been possible to check the depend- 

 ence of fluctuation on frequency by transmitting 

 signals simultaneously at two widely separated 

 frequencies and by noting the correlation between 

 their instantaneous amplitudes.' These trials indi- 

 cated a partial but significant correlation between 

 the fluctuations at two widely separated frequencies. 



To understand the significance of this result, it is 

 necessary to explain in a few words the mathematical 

 meaning of the term correlation coefficient. If there 

 are two time series, say K\, Ki, • • • , and Li, L^, ■ ■ ■ , 

 then the correlation coefficient between them is de- 

 fined (in close analogy to the self-correlation coeffi- 

 cient of one time series, introduced earlier in this 

 chapter) as the expression 



KnLn — KL „ o 



Pk,l = ,a\ = K^- K- 



(33) 



This expression equals unity if there exists a rela- 

 tionship 



L^ = aKn-\- ^,a>Q,n= 1, 2, 3, • ■ • (34) 

 or, in other words, if L is a linear function of K with 



a po.sitive slope. If a < 0, p/^^ will equal — 1 . If there 

 is some tendency of large values of L to be coupled 

 with large values of K, and small values of L to be 

 coupled with small values of K without the existence 

 of a rigorous linear relationship (34), then p^',/, will 

 have a positive value less than 1; conversely, a 

 negative value of p^-,/, (greater than — 1) will signify 

 a coupling of large values of L with small values of 

 K and vice versa. If p^.l vanishes, then the deviations 

 of individual K values from K are statistically inde- 

 pendent of (uncorrelated with) the deviations of the 

 corresponding L values from L- 



It was found that the correlation coefficient be- 

 tween simultaneous signals at two different fre- 

 quencies varied from to 0.75 with an average of 0.3. 

 In other words, while there was some tendency for 

 strong 24-kc signals to be coupled with strong 56-kc 

 signals, the simultaneous signal amplitudes at these 

 two frequencies were far from proportional to each 

 other. The same statement holds for each of the three 

 other frequency pairs at which experiments were 

 performed. It must be concluded that the observed 

 fluctuation is caused by a combination of mecha- 

 nisms, of which some operate independently of the 

 signal frequency (refraction and roll and pitch), 

 while others depend on the transmitted frequency 

 (interference) . 



7.2.4 



Summary 



The experiments carried out with a deep sound 

 source and a deep hydrophone indicate that most of 

 the observed fluctuation disappears if the whole 

 transmission path is more than 100 ft below the sur- 

 face. They also show that the surface-reflected signal 

 by itself (without interference from another path) 

 fluctuates more strongly than the composite signal 

 observed in .shallow transmission, at least when the 

 incidence at the surface is not glancing. Unfortu- 

 nately, these findings are not helpful in a choice be- 

 tween the various mechanisms which have been 

 considered. 



If roll and pitch contributed significantly to fluc- 

 tuation, its effect on a cable-supported transducer 

 would be very much less noticeable than the effect 

 on a transducer rigidly connected with the hull of the 

 ship; but there are not yet enough data with a shal- 

 low-cable transducer to permit any conclusions. 

 Image interference fluctuation will cease to operate 

 when the surface-reflected sound can be separated 

 from the direct signal. Microstructure will probably 



