TABLE I. Adjusted phase and intensity statistics at 406 Hz for data interval 6( = 5.0394 min. 

 The columns marked «>, 20, 10 correspond to the suppression of fades exceeding « (no 

 change), 20, 10 dB, respectively. A', Y, fl= CA:^+ F^)"^ in arbitrary pressure units; to cor- 

 rect to absolute level (dB/MB), subtract 169. dB for midstation and 173.0 dB for Bermuda. 



Mid station 



Bermuda 



Range from Eleuthera (km) 



Record length 



(4^^, ((&i)h (cycles^) 



F (dB) 



Number of terms replaced 



(X^) (xlO-") 

 (Y^) (xlO-") 



((6X)^) (xlO-") 

 ((6F)2> (xlQ-") 



(r2) (X10-") 



4 (dB) 



(/> (dB) 



a^>-(/)2 (dB^) 



((61)^) (dB^) 



(I 67- 6*1 ) (dB cycles) 



quency v defined by 



FAo^) = {2i,)-"'-{nyv)eM- 



V^^) 



(16) 



(10) 



is the same for every single-path signal. The root- 

 mean-square multipath signal amplitude n is defined by 



M^ = E(^?>.- ■ 



(11) 



It turns out that all important statistical properties of the 

 multipath signals are determined by the parameter v. 



III. CARTESIAN STATISTICS 



The statistical model of Sec. n predicts a covariance 

 function for the multipath signal X which is a sum of 

 contributions from the single-path components, namely, 



(X«)X(/+T)) = X<iJi>,<COS<^,.(/)cOS(/>i(/+T)>,- . (12) 



Now we assume that for each single-path the root- 

 mean-square phase fluctuations are of the order of one 

 cycle or larger, that is to say 



([*i (/)?),> (277)^. (13) 



This is not really an additional assumption but is al- 

 ready implied by (8). [The ocean model (Sec. IV) gives 

 rms (^, =877 for Bermuda. ] We are also assuming that 

 the single-path phase differences [(^,(0 -<^i(/+t)] are 

 Gaussian random variables, with a variance 



{[4>m-'t>At+T)T)r 



(14) 



according to Eq. (10). Putting together Eqs. (12), (11), 

 and (14), we find 



<X(OX(i+T)) = in'exp(-|i/^^) 



(15) 



The Fourier transform of this quantity is the spectrum 

 of X{.t), namely, 



and similarly for Fyiui). The advantage of the Car- 

 tesian spectra over the more traditional polar coordi- 

 nate representation involving intensity and phase (to be 

 discussed later) is that the Cartesian multipath and 

 singlepath spectra are simply related. It is disappoint- 

 ing that the inequality (13) applies, so that the Carte- 

 sian statistics (single path or multipath) provide only 

 such limited information about the ocean medium, 

 namely, the two parameters ix and v. (This limitation 

 would not apply at short ranges or low frequencies.) 



Figure 5 shows the Cartesian spectra with plots of 

 Eq. (16) drawn for indicated values of v'^ . An alterna- 

 tive method for estimating the value of v is to use the 

 formula 





(17) 



The values so inferred are summarized in the first two 

 lines of Table II. 



IV. INTERNAL WAVES 



These values can be compared with those derived 

 from a theory of sound propagation through a fluctuat- 

 ing stratified ocean. Starting with a spectrum of inter- 

 nal waves^ empirically derived from various oceano- 

 graphic meastirements, Munk and Zachariasen^ obtain 



{(^^,. = 8jT-2<(6C/C)i,„)92BiJa,,^oln(«ais/ajJO"'), (18) 



for a ray along the sound axis. Here n^, n^is are val- 

 ues of the buoyancy frequency at the surface and sound 

 axis, and 6 C/C is the sound velocity perturbation due 

 to internal waves, q is acoustic wavenumber, B the 

 scale depth of stratification (dn/dz = -z/B), R is range, 

 a)i„ is inertlal frequency, and j is the internal wave 



238 



