10' 



cycles per day 



10 



100 



10' 



10 

 8 



MID-STATION 



10' 



10' 



10' 



Hz 

 FIG. 7. Spectra of phase difference and intensity (bandwidth 

 0.915 cpd). The computed curves are drawn for indicated val- 

 ues of i'"^ in seconds. The area under the intensity spectra 

 (e.g., the mean-square fluctuations) is independent of i'. 



is by (2.78 iv/)"V2?T = 20-30 cycles (for comparison, see 

 Fig. 4). 



We ne.xt obtain from the random walk model pre- 

 dicted spectra for the quantities : and (p. The spectrum 

 of the Cartesian components X, Y was already given by 

 (13) in Sec. IE. 



The high-frequency spectra of i and <p are dominated 

 by the fade-outs. Each fade-out is approximated by a 

 segment of track in which the Cartesian components 

 {X, Y) move linearly, so that 



^(t) = ln[VHt-tJ' + R'] , (35) 



0(0 = arctan[K(/-0/fl] + const , (36) 



where V, R and t„ are random variables. Taking 

 Fourier transforms of Eqs. (35) and (36) and averaging 

 over the variables V, R, /„, we find the spectra 



f»(a))=ii^, (co)= 1^2^-3 (37) 



valid at high frequencies when u}»v. In performing 

 these averages we used the probability distribution of 



fade-outs given by Eqs. (22) and (24). 



The spectrum at low frequencies will be dominated 

 by the phase-wandering described by Eq. (34). The 

 Fourier transform of Eq. (34) gives 



F^(u}) = (ai'/Tr)u}-^ . (38) 



for to « 1^ . In the case of i , we expect F, (oj) to be finite 

 at low frequencies since i(0 does not wander but re- 

 mains bounded as / - «>. We know the total variance of 

 I from Eq. (20), so that 



F,(u))rfa) = (7rV6) 



(39) 



(40) 



Spectra consistent with Eqs. (37)-(39) are 



i^(co)=2l^V' + 127^-^2)-^/2 , 



where c = ^7ro"' = 0.90. Both spectra show the expected 

 transition from low- to high-frequencybehavior at o}~ v. 



We cannot exjoect this crude model to give exact quan- 

 titative information about the spectra. Accordingly we 

 modify Eqs. (38) and (37) to the form 



(41) 



(42) 



/".(a))=if,(a.) = ^xii/2^-3 , 



for low and high frequencies, respectively. A numeri 

 cal experiment (Sec. vn) can be fitted to 



a = 1.6, 3=2.0 , 

 which gives 



i^»(a)) = iv2a;-2(a,^ + 1.27i.2)-"2, 



f,(a>) = 4i.2(to= + 2.43i/V . 

 Figure 7 shows the comparison between the computed 

 spectra (42) and the observed spectra.' The overall 

 agreement is not good. The high phase values at the 

 lowest frequency band (over and above random walk) 

 could be the result of coherent modulation by large- 

 scale ocean features; some of it might be due to tides 

 (Sec. VHI). The predicted o)"' and oj"^ roUoffs for rate- 

 of -phase and intensity spectra are borne out at midsta- 

 tion. The high-frequency Bermuda intensities are 

 aliased from undersampling. 



Computed mean-square variations are 



■'0 



<'">= f" F,(a;)rfw = 2i/2[sinh-'a-a(l+a2)-i/2] ^ (43) 

 -'0 



a = (7r/2V3")a)'/i', 

 and these become logarithmically infinite as oj' — «=. 

 The upper limit is set by the integration time 6/, and 

 crudely a)'= 27r/6/. Results are given in Table IV. 



TABLE IV. Root-mean-square phases and intensities (w' 

 = 0.0208 sec-'). 



MidstaHon, ■■' = 357sec Bermuda, i'"' = 250seo 



rms 5* rms 61 rms 6* rms 6/ 



Computed (Eq. 43) 

 Observed (Table I) 



0.19 cycles 7.0 dB 

 0.17 4.9 



0.26 cycles 8.9 dB 

 0.25 6.9 



241 



