CALCULATION OF SOUND INTENSITY FROM RAY PATTERN 



55 



The formulas are' simplified in this section by using 

 various simplifying assimiptions concerning the 

 velocity-depth variation. 



Direct Beam in Linear Gradient 



Let us assume that the sound velocity increases or 

 decreases linearly with depth, with the gradient a. 

 Then, 



a = f- (72) 



dy 



Since the velocity is never constant with increasing 

 depth in this case, the approximate equations (69) 

 and (70) are applicable. Using these equations, we 

 find that the range R is given by the expression 



R = {dh — 6q) 



a 



(73) 



and that the derivative of the range with respect to 

 Ba is given by 



^ = i(l-^")- (74) 



a^o a\ ej 



Substituting these expressions into equation (71a), 

 we obtain for the transmission anomaly the expres- 

 sion 



A = 10 log 1 = 0. (75) 



The transmission anomaly vanishes, at least in this 

 approximation. If we had used the rigorous expres- 

 sions (67) and (68) for R and dR/ddo, and the exact 

 form (71) for the transmission anomaly, the following 

 formula would have been obtained, which is rigor- 

 ously correct, 



A =20 log cos do. (76) 



It may seem surprising that the transmission 

 anomaly (76) does not depend on the sharpness of the 

 velocity gradient. This seeming discrepancy results 

 from the use of the horizontal range R in the defini- 

 tion (65) of the transmission anomaly instead of the 

 slant range r. 



The results (75) and (76) for this case of uniform 

 downward refraction apply only to the sound field 

 at points actually reached by the direct rays. If the 

 water is very deep, there are portions of the ocean 

 where no sound ray penetrates, as illustrated in 

 Figure 24; in such regions the ray theory predicts a 

 vanishing sound intensity and thus an infinite trans- 

 mission anomaly. 



Reflected Beam in Linear Gradient 



We now calculate the intensity along a ray which 

 has suffered one or more bottom reflections, for the 



same linear velocity gradient assumed under "Direct 

 Beam in Linear Gradient" in Section 3.4.2. First we 

 shall assume one bottom reflection. This situation is 

 pictured in Figure 16A where the ray hits the bottom 



OCEAN BOTTOM 

 A ONE REFLECTION 



OCEAN BOTTOM 

 B SEVERAL REFLECTIONS 



Figure 16. Reflection of sound ray from sea bottom. 

 A. One reflection. B. Several reflections. 



at an angle Sj, {db > 0) and is reflected at the angle 

 —df The rays will be refracted downward, as indi- 

 cated; and the incident and reflected rays will be 

 circular arcs with equal radii. 



We can compute the horizontal range R by equa- 

 tion (69), which is valid for all cases where the ray 

 path is made up of several arcs, provided care is 

 taken in breaking up the interval of integration cor- 

 rectly. 



R 



— Co -de - Co I 



J Sa aC J 96 



'dy 

 dc 



de 



- -{e, - do) - -(9, + 9,) 



a a 



Co 



{29i + 9^- Bo), 

 a 



To use equation (71a), we must also calculate dR/d9o. 

 Using equation (74), we obtain 



aR _ Co/ 6^"\ Co/ 

 ddo a\ eJ a\ 



^co/_e,^Bj\ 



a\ 9b 9j 



Substitution of these expressions for R and dR/d9o 

 into equation (71a) gives 



10 log 



'(-^^:) 



(77) 



29b + 6h — 9o 



For the case of. a ray suffering n -\- 1 reflections, 

 pictured in Figure 16B, the procedure is similar ex- 



