radians per minute (rpm) 

 FIG. 8. Computer simulation of multipath statistics. The 

 curves labeled Fj.lu) give the average spectra of the random 

 single-path input functions <+i(f); solid: a bandlimited (2— 24 

 cpd) uniform spectrum, and dashed: a w"' spectrum above 2 

 cpd. For both cases (i>}) = l/{5 min)^. The resulting spectra 

 of the multipath Cartesian components Xit) and YU) (solid for 

 (i)", dashed for u"^) are in good accord with the predicted 

 Gaussian behavior [Eq. (16)]. At high frequencies the com- 

 puted spectra are too wiggly to be plotted; they fall within the 

 limits of the shaded band. 



VII. NUMERICAL EXPERIMENT 



Figures 8 and 9 show the results of numerical ex- 

 periments. The singlepath series 5(pi{t) were gener- 

 ated from random numbers for two cases: (1) a band- 

 limited (2-24 cpd) white spectrum and (2) an oj'^ spec- 

 trum for CD > 2 cpd (computed by accumulating random 

 S^fpi). The singlepath phase series are formed by 4>i(0 

 = Zt=o5<?i(0, and the multipath according to 



10 



X(t)='^RiCOS(pi = Rcos(l> , 



and similarly for Y{t), with R, arbitrarily set to 0. 1. 

 Spectra were computed for X, Y, <p, I. This computa- 

 tion was repeated ten times (using, of course, different 

 random noise series), and an average of the spectra so 

 obtained has been plotted. The results are essentially 



representative of MMI. The input series consist of 

 2880 terms each, interpreted as a one-day record at 

 5-min intervals. This sampling rate 6/ was chosen by 

 trial and error to avoid ambiguities in multipath phase 

 during occasional fade-outs. It would thus appear that 

 v5f= tV would give adequate sampling for a field exper- 

 iment. 



Figure 8 shows the average of the 100 input spectra, 

 and the associated Cartesian spectra according to Eq. 

 (16); these provide a check on the numerical experi- 

 ment. The spectra FXoo) and F4*(a)) = aj^f,(a)) in Fig. 

 9 have been fitted by Eqs. (42). 



VIII. TIDES 



The tidal contribution to the acoustic fluctuations has 

 been emphasized in the literature,^ perhaps because of 

 a superficial resemblance of the phase fluctuation <^(/) 

 to tidal records (Fig. 4). Our conclusion is that tides 

 play a significant but not dominant role. We shall dis- 

 cuss three hypotheses: a coherent modulation of the 

 acoustic transmission by surface tides; a coherent 

 modulation'by internal tides at the terminals; an inco- 

 herent modulation by internal tides along the entire 

 transmission path. Unfortunately, the evidence does 

 not lead to a clear-cut decision. 



For orientation we have put together an order-of- 

 magnitude summary (Table V) of amplitudes of tides 

 and internal waves (a rash extrapolation of recent com- 



the same for the cj" and oj"^ spectra of (p, (which bracket 

 the theoretical oj"' spectrum^), as expected. 



For both cases we have taken i'^ = {tp'p = 1/(5 min)^. 



10" 10' 1 



radians per minute (rpm) 

 FIG. 9. Spectra of multipath intensity and rate of phase from 

 the numerical experiment, corresponding to FJ (w)~u'' (solid) 

 and oj"^ (dashed), respectively. 



242 



