SHADOW ZONE AND DIFFRACTION 



65 



Clearly the expression B of equation (98), the re- 

 mainder term will be negligible compared with the 

 other terms if 



Xo(log.4)'«F' (99) 



XS(log^)"«(FT, (100) 



where the prime denotes any spatial derivative, and 

 <C means "is negligible compared with." If V even 

 approximately satisfies the eikonal equation (13), 

 then 



V ~ n, (101) 



where the symbol ~' signifies "is of the same order of 

 magnitude as." 



Another useful relation is obtained from equation 

 (96). The functions A and V must satisfy equation 

 (96) as long as the surface (91) has the significance 

 of a general wave front. But equation (96) implies 

 that 



V" '■^V'ilogAY, (102) 



which in turn implies that 



XoF" ~ F'Xo(log AY « V'V (103) 



because of equation (99). Combining equations (103) 

 and (101), 



Xo V" « n\ (104) 



In the ocean the index of refraction n is of the order 

 of magnitude of unity. Then, the relation (104) may 

 be stated in the following words. The first spatial de- 

 rivative of V must not change much over a spatial dis- 

 tance of one wavelength. The first spatial derivatives 

 of V give the direction of the rays; while the second 

 derivatives, yielding the rate of change of ray direc- 

 tion, give the curvature of the rays. Therefore, the 

 condition (104) becomes the following. The direction 

 of the ray must not change much over a distance of 

 one wavelength. In regions where the ray curves 

 very strongly, ray acoustics cannot be applied safely. 

 Differentiating the eikonal equation (13), we get 

 V'V" -^ nn' or V" ~' n' because of equation (101). 

 In view of equation (104), this means that 



Xon' « n" ~ 1. (105) 



In other words, the index of refraction must not 

 change much over a distance of one wavelength. 



We derive one more restriction — this time on the 

 amplitude function A. From equations (102) and 

 (104), we also have 



Xo(logA)'<l. (106) 



The relation (106) means that log A must not change 

 much over a distance of one wavelength. Since this 

 change is very nearly \oA'/A, this means that the 



percentage change in A over one wavelength must be 

 very small. 



We can summarize our conclusions as follows. The 

 eikonal equation usually will not lead to a good ap- 

 proximation (1) if the radius of curvature of the rays 

 is anywhere of the order of, or smaller than, one wave- 

 length, or (2) if the velocity of sound changes ap- 

 preciably over the distance of one wavelength, or (3) if 

 the percentage change in the amplitude function A is 

 not small over the distance of one wavelength. 



3.6.3 Comparison of Ray Intensities 

 and Rigorous Intensities 



It follows from the results of Section 2.7.3 that if 

 the general wave fronts are defined by equation (91), 

 and the instantaneous acoustic pressure by equation 

 (92), then the rigorous intensity is given by 



and, further, that the direction of energy flow is char- 

 acterized by the direction numbers dV/dx : dV/dy : 

 dV/dz. The latter direction is perpendicular to the 

 general wave front ; thus, if the wave fronts are eikonal 

 wave fronts, the energy flows along the rays in the 

 rigorous case. If the wave fronts are approximately 

 eikonal wave fronts, then the directions perpendicular 

 to these wave fronts represent very nearly the true 

 direction of energy flow. 



Thus, if the conditions for eikonal wave fronts de- 

 rived in Section 3.6.2 are satisfied, the energy ema- 

 nating from the source into all solid angles will re- 

 main within the tubular confines assumed in deriving 

 the ray intensity. We can therefore say, intuitively, 

 that if the wave fronts are very nearly eikonal wave 

 fronts, the ray intensity will be very close to the 

 rigorous intensity. Further, we can say that in both 

 cases the intensity will be given by 



I 



a' 

 2^' 



(108) 



since 



3.7 



KaF\2 /dV\ {dvy\i 



Co 



n = — • 



SHADOW ZONE AND DIFFRACTION 



When the velocity decreases from the surface down- 

 wards, the ray theory predicts a sharp shadow 

 boimdary across which no sound ray penetrates; a 

 typical ray diagram for such an instance is shown in 

 Figure 24. At the shadow boundary the ray theory 



