not entirely rigorous about his decomposition of the motion into turbulence on the one 

 hand and sound waves on the other. 



The basic exact equation for a motion in which the total pressure, density 

 and velocity are p, p and U is 



V 2 p = . 



dt 2 dxidxj 



We wish to obtain from this equation an estimate of the effect of a given field of 

 turbulence on an incident sound wave, and a perturbation procedure corresponding to 

 the different orders of scattering. One cannot simply assume, as Dr. Kraichnan seems 

 to have done, that the density and pressure fluctuations are due mainly to the incident 

 sound wave, and that the term on the right-hand side of this equation is small com- 

 pared with either of the other two terms when values of p, p, V appropriate to the 

 combined turbulence-sound field are substituted in the equation. This is not a correct 

 procedure because, as I showed with the help of some typical values, the fluctuations 

 in p and p due to the turbulence alone may be as large as those due to the incident 

 sound wave alone. For the combined turbulence-sound field, the term on the right- 

 hand side may be as large as the other two terms and one cannot begin the perturba- 

 tion procedure by neglecting the right-hand side. 



An equation whose solution has the incident sound wave as its first approxi- 

 mation can be obtained only by inserting in the above equation first the quantities 

 appropriate to the combined turbulence-sound field and second those exactly appropriate 

 to the turbulence alone, and by subtracting the two resulting equations. Only in this 

 way can the unwanted density and pressure variations arising from the turbulence alone 

 be removed from the problem. Of course, one is now left with dependent variables 

 which cannot immediately be given a physical interpretation, since they are simply the 

 increments in pressure, density and velocity that result from the incidence of the 

 sound wave on the existing turbulent motion; however, as I showed in my lecture, 

 the first approximation to this new equation does indeed describe the incident sound 

 wave, and the way is open for an investigation of sound scattering due to the turbulence 

 by the usual perturbation procedure (at least as far as the stage of single scattering). 

 In this way, it seems to me, procedure is made a little more rigorous, and a little clearer, 

 than in the two existing presentations. 



R. Kraichnan 



I think the whole idea of the decomposition into "turbulence" and "sound" is 

 intrinsically non-rigorous, and at least in the formal structure of the theory, I tried to 

 avoid that by making a rigorous decomposition into transverse and longitudinal velocity 

 fields. If you do that, and work not with the differential equation where the fact is 

 perhaps obscured that the Laplacian operator and time operator largely cancel each 

 other — but with the integral formulation of that equation and consider explicitly the 

 contribution of the homogeneous solution, you can rigorously establish a scale of order 

 of magnitude of the different terms involved. This may have been done sketchily in 

 my paper, but it can be done quite rigorously, and I think that is the fundamental way 

 to attack the problem of rigor. 



From The Floor 



I would like to throw one more question at Dr. Batchelor or any other gentle- 

 man, and that is how much formal analysis they might have done about the time 

 dependent case. 



If you want to get the broadening of a single line, you have to get into the 

 space-time correlation of the fluctuations in order to get the line broadening in the 

 scattered energy. 



Have you done anything about that? 



429 



