58 



RAY ACOUSTICS 



eters which may be calculated from the given veloc- 

 ity-depth distribution. 



The ray path in the ith layer will be an arc of a circle 

 whose radius isci/(aicosflo) according toequation (33). 

 Let the small ray element ds be inclined at the angle 6, 

 as in Figure 20. In traversing this small distance, the 



VELOCITY 



Figure 20. Ray path in ith layer. 



ray travels horizontally a distance ds cos 6. By equa- 

 tion (32), we see that ds is given by the expression 

 — Cidd/{aiCos6i); and the element of horizontal range 

 covered by the ray in a distance ds is 



Ci 



cos ddd. 



Qi cos di 



To get the horizontal range covered by the ray in its 

 entire journey through the ith layer, we must inte- 

 grate this result between 6i and 6i+i. 



Ri = 



— Ci 



Bi ai cos di 

 c 



cos Odd 



(sin Bi - sin di+i) (84) 



a, cos di 

 Co sin 6i — sin 6,+t 



cos So 



a; 



because of Snail's law (26). The results of this para- 

 graph apply without changes of sign both to layers 

 where the velocity increases with depth and to layers 

 where the velocity decreases with depth. 



The total horizontal range K from the projector to 

 the receiver is the sum of the range components in the 

 n + 1 layers. 



■V D Co ■r-v sin di — sin di+i 

 2^Ri = 2^ 



R 



(85) 



cos 6q i=o at 



where the symbol S indicates summation. By dif- 

 ferentiating both sides of equation (85) with respect 

 toflo 



sin di+i 



dR Co sin 6q -^ sin di 



ddo COS^ do i=0 



+ 



Co 



COS flo i 



E" 1 / ddi ddi+i\ 



-( COS di—- - COS di+1—— ) 



'oi=oai\ ddo odo / 



which may be written 

 dR Co sin di 

 ddo cos^ do 



'o 2=0 a A 



+ 



sm di — sin di+i 



cos di cos do ddi cos di+i cos do dd 



ddi+\ \ 



(86) 



sin do ddo sin do ddo 



For equation (86) to be usable, we must calculate 

 ddi/ddo. By Snell's law, 



cos di = - cos do. 



Co 



(87) 



By differentiating both sides of this with respect to di 

 ddi Ci sin do 

 'ddo ~ 



Co sin di 

 sin do cos di 

 cos do sin di 



(88) 



By using ddi/ddo from equation (88), and a corre- 

 sponding expression for ddt+i/ddo, the expression in 

 parentheses in equation (86) becomes 



COS^ di COS^ di+i 



sm di— sin dt+i + 



sin di sin di+i 



1 



1 



sin di sin di+i 

 Thus equation (86) becomes 



dR _ Co sin do -^ W_}_ ^^A 

 3^0 cos^ do i=oai\smdi sindi+i/ 

 _ sin 00 Y^ Ri 



COS do i=o sin di sin fl;+i 



Putting this value of dR/ddo into formula (71), and 

 noting that dk is simply d^+i, we obtain the final result 



sin 00 sin e„+i Y^ Ri 



A = 10 log 



Z^ 



;]• 



(89) 



R cos^ do i=o sin di sin di+i- 



The expression (89) is in a form well-suited for 

 practical intensity calculations. The various angles di 

 can be computed from the known velocity-depth pat- 

 tern by equation (26), and the Ri can be obtained 

 either from equation (84) or equation (36). 



Formulas for Transmission Anomalies 



In this section, the formulas obtained for the 



transmission anomaly resulting from refraction will 



be summarized. 



In the absence of a velocity gradient, or if the 



sound velocity increases or decreases linearly with 



depth below the projector, the transmission anomaly 



in the direct beam is negligible. 



If the sound velocity decreases linearly with depth 



