cycles per day 



10 100 



MID-STATION 



Hz 



FIG. 5. Spectra of the Cartesian pressure components X, Y 

 (in arbitrary units) per bandwidth 2. 58 ■ 10"* Hz. The com- 

 puted curves are drawn for indicated values of r'^ in seconds. 



mode number. An internal-wave-weighted average of 

 /' equals {j"'> = 0.44, 0.34 for a mode scale number 

 j^ = 3, 5, respectively. The values so computed are in 

 very close agreement to those inferred from the acous- 

 tic observations (Table II). A detailed calculation al- 

 lowing for the proper ray mix leads to somewhat larg- 

 er values. 



The important feature is that there are no free pa- 

 rameters in this comparison between observed and 

 computed values. We conclude that internal waves con- 

 sistent with oceanographic observations can account for 

 the measured acoustic fluctuations. Considering the 

 idealization of the ocean model the agreement is rather 

 too close. In parti cttlar, the assumed exponential strat- 

 ification and resulting canonical sound channel fail to 

 allow for the important intrusion of Mediterranean wa- 

 ter. For further detail we refer to the original paper. 



V. PHASE AND INTENSITY STATISTICS 



The observed data (for example, Figs. 1 and 2) are 

 customarily plotted in terms of intensity and phase. 

 We are therefore interested in calculating the statisti- 

 cal behavior of i and predicted by our model, these 

 quantities being related to the Cartesian amplitudes by 

 Eqs. (6)- (8). The statistical behavior of i and ip is 

 dominated by the effects of "fade outs," which are brief 

 periods during which both X and Y are small and ip is 

 rapidly changing. It is convenient to define a fade-out 

 precisely as a time interval in which 



R<(.ii , (19) 



where € is an arbitrarily chosen threshold fraction, and 

 p. is the root-mean-square value of R according to Eq. 

 (U). (The multipath intensity drop f = 20 logjo*"' dB, 

 so € = 0.1 corresponds to a 20-dB fade-out.) If the num- 

 ber II of single-path signals is large enough, we expect 

 to find the statistical behavior of the multipath fade- 

 outs to be independent of the details of the single-path 

 components. We conjecture that "large enough" means 

 only that (1) n ^ 3, and (2) no one singlepath component 

 dominates the others. The conditions appear to be am- 

 ply fulfilled*^: h = 14, 34 for midstation and Bermuda, 

 respectively, and the relative contributions among these 

 paths varies by less than a factor of two. 



We assume that n is "large enough" so that the multi- 

 path components X and Y and their rates-of-change X 

 and Y are independent Gaussian random variables. We 

 then have two numerical predictions for the behavior 

 of I and (j). The statistical variance of i is 



(,2)-(,)2 = „V6 , (20) 



and so rms /=n^6""^ 10/lnl0= 5. 57 dB as compared to 

 the measured values 6. 2 and 5, 6 dB for midstation and 

 Bermuda, respectively. The correlation between the 

 rates of change of i and 4> is 



(U"ll(fil) 2 



((;2>(02))l/2-7r' 



as compared to 0.66 and 0.68 for midstation and Ber- 

 muda. The relations (20) and (21) are independent of 

 the details of the fade-outs, but to obtain further infor- 

 mation about the behavior of < and (p we must examine 

 the fade-outs more closely. The model predicts the 

 following statistical properties of fade-outs. 



C- 



= 0.63 



(21) 



(1) The fraction of time occupied by fade-outs is 



(2) The average duration of a fade-out is 



(3) The average interval between fade-outs is 



(22) 



(23) 



(24) 



To form an easily visualizable picture of the fade-out 

 process, we suppose that the signal components {X, Y) 

 drift past zero at imiform speed during the fade-out in- 

 terval. For this uniform-drift picture to be approxi- 

 mately valid, we require that the change in the speed 



TABLE II. Comparison between mea- 

 sured and computed values of v = rms 



1' (sec"') 

 Midstation Bermuda 



Acoustic measurements (MIMI) 



2.8x10"^ 4.0x10-3 

 2.8x10"^ 3.7x10"' 



Fig. 5 

 Eq. (17) 



Theoi-y based on internal wave model* 



Eq. (18) for;« = 3 2.9x10"' 4.4x10"' 

 Ray mix for;* = 3 3.5x10"' 5.2x10"' 



239 



