ideas have been evolved. What I have tried to do is to present reasonably reliable 

 analysis of the process of single-scattering of sound waves in some important cases. 

 It took me a long time to make sense out of the large and confused literature on 

 the problem, and I hope that the account I have given will save other people from 

 having to spend the same amount of time. 



REFERENCES 



D. I. Bhokhintzev 1945 "Propagation of sound in turbulent flow." C. R. Acad. Sci. URSS 46, 

 136. See also "Acoustics of a non-homogeneous moving medium," 1946, translated from 

 Russian and issued as N.A.C.A. Tech. Mem. 1399, 1956. 



T. H. Ellison 1951 "The propagation of sound waves through a medium with very small vari- 

 ations in refractive index." J. Atmos. Terres. Phys. 2, 14. 



R. H. Kraichnan 1953 "Scattering of sound in a turbulent medium." J. Acous. Soc. Amer. 25. 

 1096. 



M. J. Lighthill 1953 "On the energy scattered from the interaction of turbulence with sound 

 or shock waves." Proc. Camb. Phil. Soc. 49, 531. 



D. Mintzer 1953 "Wave propagation in a randomly inhomogeneous medium." I. J. Acous. 

 Soc. Amer. 25, 922; II. Ibid 1953, 25, 1107; III. Ibid 1954, 26, 186. 



A. M. Obukhoff 1943 "On propagation of sound waves in eddying flow." C. R. Acad. Sci. 

 URSS 39, 46. 



A. B. Wood 1932 "A Text-book of Sound." London: Bell. 



DISCUSSION 

 D. Mintzer 



I would like to make a few comments about a possible new experimental tech- 

 nique which may be useful in analyzing turbulence. As shown in Dr. Batchelor's 

 paper, and in previous published works, we now have a good understanding of the 

 scattering of sound waves by turbulence. This leads me to believe that we should 

 start thinking of sound scattering as a technique for determining correlation functions 

 in media which are randomly inhomogeneous. It has been shown from scattering 

 experiments that one can determine values for some of the variables of interest, such 

 as the scale of turbulence (and characteristic times, in time-varying problems). This 

 may turn out to be a very useful technique, especially in large-scale turbulence, such 

 as in the ocean and the atmosphere. 



As a simple example, you can show that by taking the differential scattering 

 cross-sections in two directions close to each other, you can obtain information about 

 the derivatives of the energy spectrum of the turbulence. 



R. H. Kraichnan 



There are two subjects I should like to discuss briefly in connection with Dr. 

 Batchelor's interesting talk: the domain of validity of sound-turbulence scattering theory 

 based on the first Born approximation, and the effect on the scattering process of the 

 time-dependence of the turbulence field. 



As Dr. Batchelor pointed out, the errors implied in the first Born (single- 

 scattering) approximation are not wholly separable from the errors involved in arriving 

 at the usual isentropic inhomogeneous wave-equation for the scattering of sound by 

 turbulence. The equations of conservation of mass and momentum do not alone pro- 

 vide a complete description of the dynamics of a compressible fluid; they must be 

 supplemented by an expression for the internal energy function of the fluid and a law 

 of entropy flow and production. A general consequence is a multi-valued relationship 

 between density and pressure in the fluid. If one makes the usual adiabatic approxi- 



423 



