/'(k)=| 



rf^xexp(-tk. x)p{x) 



(18) 



The supereikonal approximation now consists of ne- 

 glecting all momentum transfer correlations in the per- 

 turbation series. That is, we approximate (k- ki 



-ks, k„f-q^ + i€ by )?-2k- (yfe, + *2 + . . . + k„) 



+ fef + i^ + • • • + )?„- q^ + U, and neglect all terms of the 

 form k, • icj when i*j. Note that the first approximation 

 occurs in the second-order term in V. Once this sim- 

 plification is made, the perturbation series can be 

 summed exactly, and one obtains the result 



pG) 



(- 



Srr'^ 



Jo W'^YK 



W 



43 



+ 3/((3, x) + i€ 



»■ 



where 



m,x) = ^~iV{k)^ rfsexp[+i(sk-x + 3s(l-s);fe^)] 



(19) 



(20) 



This expression constitutes the supereikonal approxima- 

 tion to the pressure. The conditions under which it is 

 valid are 



qx»l, qL»l , 



and 



x«(l/q^L)(C/6Cf 



(21) 



(the last condition may in fact be too stringent). 

 Here L is the correlation length of the sound speed 

 fluctuations— i. e. , the correlation function p(x-y) 

 = ( V(x)V(y))/iq* vanishes when Ix - y I i L. 



It is worth noting that if in Eq. (20) the 0s(l - s)l^ 

 term is omitted from the exponent, we obtain 



p{x)- 



At!X 



exp 



['K^^^Io 



V{sl) lis) 1 , 



(22) 



which is the conventional WKB, or eikonal, or geomet- 

 rical optics, approximation to the pressure. The pri- 

 mary virtue of the supereikonal form, therefore, is 

 that it contains as limiting cases hotli the conventional 

 eikonal and the complete first-order perturbation-theory 

 approximations. 



While Eqs. (14) and (20) do constitute a closed-form 

 solution for the pressure, the expressions are still a bit 

 unwieldy, and further simplification is useful. To this 

 end, let us evaluate the integral in Eq. (19) by station- 

 ary phase, keeping in mind that x and q are both large. 

 The stationary phase point is /Sg, where 



Q - 



W 



i{0^) + ^a-r^I(^,x) 





From Eq. (20), we may estimate that 



^o^Aftx) 



So L' C ^° • 



Hence, if 



x< qL^C/bC , (23) 



the stationary phase point is accurately given by the 

 solution of the simpler equation 



and is located at /3o = x/2q. Thus we find 

 />(x) = ^^2(i2^exp[(,V29)/(x/29,;)] . 



(24) 



To approximately evaluate the integral / in this expres- 

 sion, we return to Eq. (19) and now expand in powers 

 of the potential V in order to obtain the first-order con- 

 tribution to the pressure: 



X \i^ log/(ft J)] . 



<^'^^4^--^)] 



We may evaluate this integral by stationary phase as 

 well; under the condition (23) the stationary phase point 

 is again at ^^ = x/lq, and we obtain, in analogy to (24), 

 the expression 



'.S'-t^^tl, -'(§-)] 



Ar\x 

 But we also know that 



Px 



(x)=[ <f^'C(:k-x')v(k!)G(^') . 



Thus we may eliminate I{x/2q, x) from (24) to obtain 



p{x) = G{x)exp[x{x)], (25) 



where 



X{x) = ~ { d'x'GG--x')V(l')G{x') . 

 G(x) J 



(26) 



This is known as Rytov's approximation to the pressure. 

 A direct derivation of it may be made by replacing the 

 wave equation (10) by an equation for log[^(x)/G(x)] and 

 solving this to first order in V.* However the justifica- 

 tion for the approximation is somewhat obscure in this 

 direct derivation; in the approach via the supereikonal 

 technique what is being left out is more clearly visual- 

 ized. 



In any event, depending on the validity of the criterion 

 [Eq. (23)1, °n6 "^^Y "^^ either the superikonal expres- 

 sion [Eq. (19)] or the Rytov expression [Eqs. (25) and 

 (26)] to proceed further. We shall use Eqs. (25) and 

 (26). 



The Green's function G is given by Eq. (14); hence 

 we may write for the quantity X the expression 



X( 



477 J 



ylx-yl 



exp[iq{ >' + I x - y I - r)] I^(y ) 



(27) 



Let us first comment on the geometrical optics limit 

 of this expression. This limit results from an evalua- 

 tion of X{x) by the method of stationary phase. Provided 

 that the Fresnel condition 



x< qL^ 



is met, the stationary phase path in Eq. (27) is the 

 straight line joining to x, and the stationary phase 

 value of ^x) is just 



Xix, 0, 0) = ^ r dx' V(x', 0, 0) , (28) 



2? Jo 



which is immediately recognized as the correct geo- 



215 



