(henceforth GM72 and GM75) with subsequent modifica- 

 tions.^ The model is contrived, and not the result of the 

 kind of the three-dimensional array measurements that 

 are really needed. Information with respect to the high 

 wave- number cutoff is particularly lacking. Still, there 

 is sufficient evidence now to make the present exer- 

 cise rewarding. And where evidence is lacking it points 

 toward the inverse method of using acoustic observa- 

 tions to improve the description of internal waves. 



We shall take an exponential stratification scale of 

 1 km. This same ocean model underlies both the in- 

 homogeneity of the sound field [e. g. , the "canonical 

 sound channel C{z)"] and the anisotropy of the 6C fluc- 

 tuations (the ratio of vertical to horizontal scale is 

 typically 1 : 10). The solutions are now in simple form 

 and require at most a single numerical integration; 

 even this can be avoided in most applications by evaluat- 

 ing the integral near the ray apex. 



Objections will be raised to the application of a model 

 ocean. There are, of course, large geographic varia- 

 tions of the water column (acoustic experiments invari- 

 ably fall into "anomalous" regions). Our position is 

 that the geographic factor in sound transmission needs 

 to be (and is in fact) taken seriously; but what is even 

 more needed are explicit solutions that permit compari- 

 son with experiment and provide an insight into the role 

 played by various ocean parameters, provided the un- 

 derlying model, though idealized, has the fundamental 

 properties of the world oceans. 



I. HOMOGENEOUS OCEAN 



The problem is to evaluate the pressure 



Re[p(x) expi- i<Jt)] (1) 



at a point x produced by a point source at the origin of 

 frequency a and wave number q (for convenience we 

 take the source to have unit strength). We then write 



p = exp{iqx)(pi + ipz) = I p| exp(i4>) = pae" , 



where /)o(x) = exp(iijlxl )/4;rlxl = exp(i9x)/4iiA- is the re- 

 corded pressure amplitude in the absence of any fluc- 

 tuations, but allowing for geometric spreading. Hence 

 (t> = X2 is phase, and 



' = log.|/.| = = <i>+2.Y, , <0 = logJ/.=„| (3) 



is intensity (multiply by lO/log^lO to obtain dB; from 

 now on we write log for log,). 



We assume the 6C fluctuations, and hence within the 

 approximation we shall use, the A' fluctuations, to be 

 Gaussian, sothat<X> = 0, <e*> = exp(5{X^». Some of 

 the interesting observables are given by 



(L')^{{iog\p\'f) = {L)' + 4{Xl), (4) 



rHiog<|/,|2> = <i> + 2<Xf>, (5) 



log|(/,>|2 = <i>+(Xf>-<Al>, (6) 



<(/)^> = (;f|), {<t>L) = 2{X,X,), (7) 



(i) = i\Po\' [exp(2<A^ » + exp(2<X*^ » 



t2exp(2(j!i?»], 



(8) 



(2) 



where 



4«|x|'>±Re<.\:^», (XiX2> = ilm(X^>. (9) 



Let us begin our discussion by neglecting the effects 

 of the sound channel. The problem of sound propagation 

 in the presence of fluctuations superimposed on a homo- 

 geneous isotropic background is easier to set up and to 

 visualize than the problem involving an inhomogeneous 

 background, so it is conceptually advantageous to work 

 out this case first. Later, when the inhomogeneous 

 background representing the sound channel is introduced, 

 the analysis can be carried out very much as in the 

 homogeneous case, and the resulting formulae, while 

 geometrically more complex, are entirely analogous 

 to those obtained in the simpler example. 



Our analysis will be based on the supereikonal ap- 

 proximation, and it will be convenient at this point to 

 give a brief review of an earlier report of this method. ' 

 The sound propagation from the source to the point x 

 is through an isotropic homogeneous ocean, in which 

 the sound speed is C, on which is superimposed a fluc- 

 tuation in sound speed 6C(x) which is, of course, very 

 small compared to C. Mathematically, then, the 

 pressure satisfies the wave equation 



{V^ + q')p{x) = V{x)p(x), 



(10) 



where q=o/C for a source emitting sound of frequency 

 CT, and where 



F(x) = 29^5C(x)/C 



(11) 



[For the inhomogeneous case we shall simply replace 

 C by C(x), and, accordingly, q by o/cO-). ] The bound- 

 ary condition associated with Eq. (10) is that as x— 0, 



/)(x)-l/47rr . (12) 



Equation (10) may be cast into integral form through 

 the use of the outgoing wave Green's function 



(v2 + ^)G(x-y) = 5'(x-7); (13) 



explicitly, we have 



G(x) = exp(i^x)/477r . (14) 



Then we may write, in place of Eq. (10), 



/.(x) = G(x)+j rf'yG(;-y)7(y)^(y) 



(15) 



Iteration of this integral equation generates the pertur- 

 bation series for />(x), which is more conveniently writ- 

 ten in Fourier-transformed form as follows: 



/.(k) = G(k) + G(k) j 1^ F(ki)G(k - k,) 



-G(k) 



J I2^J (277 



V'(k,)G(k-k, 



xF(k2)G(k-k, -k^) + •■• 

 where 



G(k) = (*'2 -?' + !«)■' , 

 and, of course, where 



(16) 



(17) 



214 



