II-9 



The theoretical problems encountered in this task are twofold: 



1. It is necessary to characterize the ensemble of micro- structures 

 which represent the slowly changing details of the medium. 



2. We must find some means for evaluating explicitly the averages of 

 the field over this ensemble. 



The micro- structure of the ocean consists essentially of patches whose 

 local sound velocity is somewhat higher or lower than the average. The most ap- 

 propriate way of characterizing this structure is by giving the spectrum of the 

 index of refraction. This spectrum is a measure of the relative number of patches 

 of any given size. From a knowledge of the spectrum we may derive such measur- 

 able properties of the medium as the correlation function of the index of refraction 

 at two different points, or the mean square difference of the index of refraction at 

 two separate points. Until the last few years, it was usual to employ an empirical 

 spectral correlation function which had little theoretical justification but made it 

 convenient to evaluate the averages over the ensemble required under step 2 above. 

 Recently, some of the work has begun to make use of our knowledge of the turbu- 

 lent mechanism which generates the ocean micro- structure, thus basing the scat- 

 tering theory on spectra which are physically justified. 



The evaluation of averages over the ensemble leads to analytical dif- 

 ficulties which usually require mathematical approximation. Giving a physical 

 meaning to the results, we may distinguish two cases, as shown in Figure II-2. 

 Consider a number of neighboring patches with an effective radius R scattering 

 an incident wave of wave number k. For any one patch, the magnitude of the scat- 

 tered wave is small compared to the incident wave. Furthermore, if the wave- 

 length of the incident wave (-j— ) is much less than the size of the scatterer R, 



the scattered wave can be shown to be highly collimated (see Section IV- B). In 

 this case, the scattered wave consists either of a slightly divergent shadow zone 

 or a slightly convergent zone of somewhat higher intensity, depending on whether 

 the local index of refraction is larger or smaller than unity. Thus the scattered 

 beam is essentially in the shape of a cone subtending a conical angle which may 



be shown to be of the order of t-=-. It is clear that this angle will be very much 



kK 



smaller than unity as long as the incoming wavelength satisfies the condition 

 kR > > 1 mentioned above. Otherwise, if the incoming wavelength becomes of the 

 same order of magnitude or greater than the size of the scatterer, the scattered 

 sound has an omnidirectional nature. We see from Figure II-2 that if the fre- 

 quency of the incident sound is sufficiently high, the "focusing" distance of the 

 patch for a parallel beam of sound would be of the order of Lq ^^ kR . For a 

 range L substantially less than L^ the principal effect of the inhomogeneities is 

 therefore the focusing or defocusing of the sound. On the other hand, for a range 

 L much larger than Lq the scattered waves from the different inhomogeneities 

 overlap, and interference effects dominate. This range is therefore called the 

 interference range. 



Arthur 21.Ilittlc.3nt. 



S-7001-0307 



