CAUSES OF FLUCTUATION 



167 



imum at twice the period, and so on, these con- 

 secutive minima and maxima becoming gradually 

 more shallow until the self-correlation coefficient 

 effectively approaches zero. In other words, hidden 

 periodicities are revealed by the location of maxima 

 and minima of the self-correlation vs interval curve." 

 It was believed, at one time, that part of the ob- 

 served signal fluctuation could be explained as train- 

 ing errors due to the roll and pitch of the transmitting 

 vessel. Self-correlation coefficients were studied pri- 

 marily with a view toward finding the fluctuation 

 periodicities which would coincide with the known 

 periodicity of the vessel. Although the results of these 

 studies were at first disappointing, it is possible that 

 future work will lead to more positive results. 



7.1.4 Space Pattern of Fluctuation 



In order to discover to what extent the observed 

 fluctuation varies in space, an analysis was made of 

 the fluctuations observed in the simultaneous out- 

 puts of two hydrophones.- The two hydrophones 

 were kept either at the same or at two different re- 

 corded depths. No determination of their horizontal 

 separation was made, but they are believed to have 

 had a horizontal separation of between 5 and 25 ft. 

 Thus, both hydrophones were at about the same 

 distance from the projector, which emitted 24-kc 

 signals. The number of samples analyzed is too small 

 to establish any quantitative law, but indications are 

 that the correlation between the simultaneous out- 

 puts tends to become weaker as the distance of the 

 two hydrophones from each other or as their joint 

 distance from the sound source is increased. How- 

 ever, in the majority of samples analyzed, the correla- 

 tion remains significant, even at the maximum verti- 

 cal separation of the two hydrophones, which was 

 300 ft.^ 



7.2 



CAUSES OF FLUCTUATION 



" This property of the self-correlation coefficient is incor- 

 porated in a mathematical theorem frequently quoted as 

 Khintchine's theorem, which states that the coefficient of self- 

 correlation is the Fourier transform of the (normalized) 

 squared frequency spectrum of the time sequence considered. 

 If, as in the case of the pendulum, the time sequence has a 

 tendency to repeat its functional pattern, its spectrum will 

 have a maximum at that frequency. This peak in the spectrum 

 of the time sequence may remain undiscovered if the time 

 sequence is inspected directly because of the changing phase 

 relations. The squared spectrum, however, contains the abso- 

 lute values of the squared frequency amplitudes, without 

 regard for phase relationships. Consequently, its Fourier 

 transform, the self-correlation coefficient, reveals the "hidden 

 periodicities" more clearly than^the original time sequence. 



' With one hydrophone at 16 and the other at 300-ft depth, 

 the correlation coefficient was 0.34 at a range of 950 yd and 



It has not yet been po.ssible to develop a theory of 

 fluctuation which would permit the prediction of its 

 magnitude and time rate as functions of oceano- 

 graphic or other parameters. Nevertheless, it is of 

 interest to consider the various mechanisms which 

 have been considered responsible for fluctuation. 

 These mechanisms may be described under three 

 headings: roll and pitch of the vessel, interference 

 mechanisms, and thermal microstructure of the 

 ocean. 



7.2.1 Roll and Pitch of Transmitting 

 Vessel 



Except in very calm weather, the transmitting 

 vessel is subject to considerable roll and pitch, with 

 the result that the bearing of the transmitter relative 

 to the target and relative to the surface of the sea is 

 not constant. Because of the directivity of the trans- 

 mitted sound beam, a change in bearing will bring 

 about a change in received signal intensity if the 

 change exceeds a few degrees. This change may come 

 about merely because the principal beam may miss 

 the target during one phase of the roll and hit it 

 during another phase. A more involved hypothesis 

 considers the interference between direct and surface- 

 reflected soimd. In the presence of a slight upward 

 refraction, the direct and surface-reflected rays to the 

 target leave the projector at appreciably different 

 angles. The sound received at the hydrophone is the 

 result of interference between these two rays. If the 

 training of the projector is changed slightly, the rela- 

 tive intensity of the two rays will also change, as the 

 projector will discriminate first against one and then 

 against the other. If the two rays are out of phase by 

 nearly 180 degrees, the resultant change in the in- 

 tensity distribution of the interference pattern may 

 become very appreciable, even with comparatively 

 minor changes in the relative intensity of the two 

 component rays. 



The chief argument in favor of roll and pitch as a 

 cause of signal fluctuation was that in the earlier 

 studies the self-correlation coefficient seemed to indi- 

 cate a period of fluctuation similar to the known 

 period of roll of the transmitting ship. Subsequent 



0.38 at a range of 1,750 yd. The number of signals in each 

 sample was 40. For a definition of correlation coefficient, see 

 Section 7.2.3. 



