50 



RAY ACOUSTICS 



where Co is the velocity of sound at the projector. 

 Since the actual ray path is curved, all we can say is 

 that y is some function of the surface velocity, the 

 apparent angle do, and the velocity-depth pattern. 

 We can, however, give an exact expression for the 

 increase in y during the time interval dt. If c is the 

 sound velocity at the depth y, and 6 is the inclination 

 of the ray at the depth y, we have 



dy = c sin ddt. (39) 



We now take differentials of both sides of equation 

 (38), obtaining 



dyo = Co sin dodt. (40) 



By dividing equation (40) by equation (39) to elimi- 

 nate the time, 



dyo Co sin do 



dy c sin 6 



(41) 



We eliminate d from equation (41) by using Snell's 

 law (26) : 



sin 6 = Vl - cos^e = V 1 - [(c/co) cos OoJ 



so that equation (41) becomes 



dyo sin Oo 



dy 



^|/i-(^cos.oy 



(42) 



The quantity c/co represents the variation of velocity 

 with depth. If « is defined by the relationship 



Co 



then € represents the relative change in velocity as a 

 function of depth. Rewriting equation (42) in terms 

 of €, we obtain 



dyo _ sin ^o 



d^ ~ (1+6)[1 - (1 -I- 6)2 cos2 eo]= 



= { (1 -h if csc2 doll - {1 + €)2 cos^ ^o] } ~' (43) 

 = [1-1-2(1- cot2 (9o) e -h (1 - 5 cot^ do) t- 

 - 4 cot^ do e^ - cot- 00 e"]"- 



upon multiplying out and collecting terms. 



Since percentage changes of sound velocity are 

 always small in the sea, the quantity e is a very small 

 fraction, almost always less than 0.02. Consequently, 

 the terms with e^ or higher powers of e in equation 

 (43) may safely be neglected, giving approximately 



^° = [1 + 2efl - cot^ 0o)]--. 

 dy 



If we define w by 



w = 2(1 - cot^^o). 



(44) 



equation (44) may conveniently be rewritten as 



dy 



dyo = ,, , ,■ • (45) 



(1 + we)' 



It may be noted that although t is always much less 

 than one, we is not necessarily so. We now integrate 

 both sides of equation (45) between and the true 

 depth Y, obtaining 



r dy 



Jo (1 -[- we)'- 



(46) 



The expression (46) provides a functional relation- 

 ship between the true depth Y, the apparent depth 

 Yo, and the velocity-depth variation e{y). In any 

 practical situation it is possible to determine Yo and 

 do, and €{y) can be deduced from the temperature- 

 depth variation indicated by the bathythermograph 

 slide. Thus, all quantities in equation (46) are known 

 except the true depth Y, which occurs only as the 

 upper limit of integration. The value of Y may be 

 estimated by using trial values for the upper Umit 

 of integration and by seeing which trial value yields 

 a value for the definite integral closest to the known 

 left-hand side Yo. If the velocity-depth variation is 

 not simple, the integrals must be evaluated by numer- 

 ical integrations; but if e(y) is a linear function of y, 

 or a succession of linear functions of y, the integrals 

 can easily be evaluated exactly. 



Tables have been developed by the use of such 

 methods for the depth errors expected in the presence 

 of various types of velocity gradients. In preparing 

 these tables it was assumed that the sound velocity 

 versus depth curve could be approximated by judi- 

 ciously chosen straight-line, segments without intro- 

 ducing too much error in the calculated depth error. 



Though equation (46) is the relation used in the 

 construction of depth correction tables, it is interest- 

 ing to carry the approximation two steps further. If 

 we expand the integrand in powers of we and neglect 

 all but the first two terms, we get 



Fo = J^ (1 -iwe+ ■■■)dy 



which becomes 



Y — Yo = iw I edy + terms in (weY. 



To the same order of approximation, Yo may be sub- 

 stituted in place of Fas the upper limit of integration, 

 which gives 



e dy + terms in (we^. (47) 



