is much smaller than the term with A]\. This is be- 

 cause, as we shall see below, the spectrum F((jj,j;z) 

 tends to weigh small values of o) much more heavily. 

 Thus for most purposes we can ignore this term, and 

 replace Eq. (74) by the much simpler expression 



{X^) = -2q^,r'\ioB { dxsec^eJ2 t exp(i/3/) 



r 



F(u>,j;z) 



(O) - OJr) 



(75) 



(76) 



where 



& = (Tr/Bf{n/nofl/qA„. 



Using the special expressions for A,^ derived above ap 

 plicable to specific regimes; we can write, for single 

 loop downward rays, 



^'^"^^r(i^-x) 



/3 = /3s 



■a)© 



qR 



for near axis rays 



\2 



^^-^©(S)i 



sinKrsinA'(fl- x) 

 Kq sinKR ' 



and for the long-range many- loop situation, 



^^ '"\^x{R-x) tan^e 



(3 = ^^ 



■©er 



qR 



Equations (66) and (75) are as far as we can go toward 

 computing the quantities of interest without committing 

 ourselves to a particular spectrum F, and a particular 

 sound channel and associated ray paths. It therefore 

 now becomes necessary to turn to a discussion of these, 

 and Sees. IV-VI will be devoted thereto. 



IV. CANONICAL SOUND CHANNEL 



Let T(z), S(z), P(z) designate the undisturbed distri- 

 butions of temperature, salinity, and pressure. The 

 velocity of sound is a known empirical function of these 

 variables, C(z)= C(T, S, P), having typically a minimum 

 value C = C at some depth z = - h and increasing by a few 

 percent towards top and bottom. The (fractional) veloc- 

 ity gradient can be written 



C"'3.C=a- 9.r+/3- a.s + y a,p , 



(77) 



with 



(a, (3, v) = C"' • {dT,3s,»p)C . 



The temperature gradient is the sum of potential and 

 adiabatic gradients, S,T = B,Tp + a^Tji , so that 



C-*9,C^=a- 9,r^+y 3,P=(-0. 03-l.ll)xlO"= km^' 



= -1.14X10"^ km "' = ->-_, (78) 



is the fractional velocity gradient in an adiabatic iso- 

 haline ocean. In analogy, we define a potential velocity 

 gradient such that 



and write the potential gradient in terms of the buoyancy 

 frequency n(z): 



c-'a,c= (iJ./g)nH2)-yA, 



(79) 



nHz) = - gp-^a,Pp - giaa.Tp - hd,S) = ga3 ,Tp{l - Tu) , 



(80) 

 where the "Turner number 



63,5 



Tu- 



aS.Tp 



gives the relative contributions of salt and (potential) 

 temperature to the potential density stratification. In 

 Eq. (79) 



p. = (a/a)s(Tu) , a/a = ZA.%, 



s(Tu) = (\ + cTu)/(l- Tu) , c = a/3/a6 = 0. 049 , 

 using the numerical values 



a = 3. 19x10"' (°C)-' , = 0.13x10-' ("C)"^ , 

 /3 = 0.96 V 10"' (%o)"' 

 y = l. 11x10-2 km-* 



6 = 0. 80^10-' (%„)■' , 



The a value is typical of conditions in the sound channel 

 (it may vary by as much as 50% between surface and 

 bottom). In shallow water the )i^ term dominates, and 

 the velocity increases upwards; in deep water n^~0 and 

 the velocity increases downwards at the rate y^. At the 

 axis of the sound channel 3,C = 0, hence 



„{!=- h)^n= igyjH)"' . (81) 



An exponential stratification model 



n = noe''^ , B = 1 km , 



Wo = 5. 2 X 10-' sec-' («„ = 3 cph) , ^^^' 



gives a reasonable fit to the oceans beneath the thermo- 

 cline' (we ignore the surface mixed layer and interpret 

 rio as a surface extrapolated value). The sound axis is 

 at a depth 



- z=/z = Blog(no/«) = 0. 89 km + jBlogs , 



(83) 



compared to typically observed values 0. 7-1. 5 km. 

 Geographic variations in the sound axis are associated 

 with the temperature dependence of n through the a pa- 

 ramet^er, and with the salinity dependence of s(Tu). We 

 take h=l km. In terms of a dimensionless distance r] 

 above the sound axis, the velocity profile can be simply 

 written 



C-C[l+€(e''-7;-l)l , €=iBr^ = 5.7x10-', 



■n = (z- D/^B . 



(84) 



The coefficient e is readily interpreted as the fractional 

 adiabatic velocity increase over a scale depth. Equa- 

 tion (84) is a reasonable description of an oceanic sound 

 channel (Fig. 1), given in terms of physical constants 

 of seawater and the stratification parameters hq, B, Tu. 

 We require certain geometric properties of the sound 

 channel. Let z(x) denote a ray with inclination dz/dx 

 = tane and curvature d^z/d/ ^sec^6- d9/dx. From 

 Snell's law cos0 = C/C, where C is the velocity at the 

 ray apex. Hence 



(R- 



dH 



de 



2e C 



= — 5- = sec^6>- — tan6l = -f- -^ sec9(l - e") = y. (1 - e 

 a.ir dz B C 



•). 

 (85) 



The range of a double loop is 



221 



