56 



RAY ACOUSTICS 



cept that n complete joiimeys from the bottom back 

 to the bottom must be added to the mterval of 

 integration. The calculated transmission anomaly for 

 a ray which leaves the source at the angle do, suffers 

 n + 1 bottom reflections, and strikes a receiving 

 hydrophone at the inclination 0k, turns out to be 



A = 10 log 



[2(n +l)-l+ '' 



Bh 



2(w +1)9^ + 0^- 00 



(78) 



Layer Effect 



When sound originates in an isovelocity layer or in 

 a layer with a weak velocity gradient and then passes 

 into a layer with a sharp negative gradient, the sharp 

 refraction results in an extra spreading of the sound 

 rays and a consequent drop in intensity. This phe- 

 nomenon is called layer effect, and is of operational 

 importance. We shall consider only rays which leave 

 the projector in a downward direction, so that the 

 formula (66) for the transmission anomaly will apply. 



Two separate cases will be treated. First, we shall 

 consider the velocity-depth pattern shown in Figure 

 17: an isovelocity layer, followed by a layer of 



VELOCITY^ 



Cq-AC 



SOURCE 



HYDROPHONE 



FiGUKE 17. Bending of ray by temperature discon- 

 tinuity. 



negligible thickness with a very sharp gradient and 

 a total drop of soimd velocity of amount Ac, followed 

 in turn by a second isovelocity layer with the velocity 

 Co — Ac. If the ray direction in the first isovelocity 

 layer is 0o, and in the second isovelocity layer 0k, we 

 have by Snell's law (26) 



cos dh 

 cos 00 



Co 



Ac 



= 1 - 



Co 



Ac 



Co 



If the angle 9 is small, we may replace its cosine by 

 its approximate equivalent 1 — S^/2. Using this ap- 

 proximation, and dropping the negligible term 

 (Ac/co)^, the preceding equation becomes 



' Co 



(79) 



If hi is the height of the soxmd source above the 

 abrupt velocity change, and ^2 is the depth of the 

 hydrophone below the velocity change, we easily find 

 from formula (66) that 



A = 



+ 10 log 00 + 10 log Oh 



= -»{l|(' + t1)]' 



(80) 



smce 



„ ^1 ^ 



R ^ - +-■ 



00 0h 



Next we shall consider the velocity-depth pattern 

 shown in Figure 18: an isovelocity layer extending to 



GRADIENT 



FiGOBE 18. Bending of ray by deep thermocline. 



a depth hi below the sound soiu-ce, followed by a layer 

 of indefinite extent with the constant velocity gradi- 

 ent —a. At a depth y' below the top of the gradient 

 layer the soimd velocity will be Co -)- ay'. We there- 

 fore obtain the following expression analogous to 

 expression (79) for the ray direction 0{y') at the 

 depth y'. 



0{y') = V^ - 2{a/co)y'. (81) 



We shall use equation (66) to calculate the intensity 

 of the sound received by a hydrophone at a depth h^ 

 below the top of the gradient layer. Since the ray 

 direction in the isovelocity layer is constant at 0a, the 

 separate integrals in equation (66) have the values 



dy' 



Jo 00 Jo Vfo - 



2{a/co)y' 



The last term may be integrated directly, and with 

 use of equation (81) 



{00 - 0h) 



f'dy ^hi 



Jo 00 



a/co 



^0 K^O + 0h) 



