where a is the velocity of sound, k the radian wave-number of the incident wave, k 

 that of the turbulence, and E(k) its three-dimensional energy spectrum. 



G. K. Batchelor 



I think that Professor Lighthill's remarks about double and multiple scattering 

 take it for granted that an investigation by the usual iterative procedure is permissible. 

 In the particular case of sound waves scattered by turbulent motion of the medium, it 

 is not yet proven that this procedure is valid beyond the stage of single scattering. To 

 say again in one sentence what I said in the lecture, if one takes the quantities appro- 

 priate to the scattered wave, and inserts them into the right-hand side of the basic equa- 

 tion, one may obtain terms that are of the same order of magnitude as those that have 

 already been neglected in deriving the equation; in this way, an inconsistency may arise. 



R. Kraichnan 



One of the problems may be that it is a very broad problem, trying to decide 

 which higher order effects are important where. Take the specific problem Lighthill 

 brought up: a region which receives no single scattering at all. Then it is quite 

 possibly safe to use the double scattering approximation to find out what happens there. 

 But if you consider a region which can receive both multiple and single scattering, and 

 where the scattered energy accumulates from large volumes, you must take account of 

 slight deviations from single scattering results, and this requires taking into account 

 neglected terms in the differential equation. Then, there are a lot of high order terms 

 that come in. 



It is extremely complicated to separate all of these high order effects. 



G. K. Batchelor 



May I add a word concerning Professor Lighthill's remark earlier that he 

 wasn't quite clear why I preferred my derivation to the ones that he and Dr. Kraichnan 

 have given? I am sure this needs more informal argument over a table, rather than 

 discussion here, but I can perhaps give a qualitative description of the reason. 



There are really two reasons. The more obvious one concerns Professor Light- 

 hill's presentation, and arises from the history of his work on scattering. His work 

 was done when he was in full pursuit of his theory of the generation of aerodynamic 

 noise, which, as everybody now recognizes, has made a fundamental advance in our 

 understanding of that problem. An essential and important part of that theory is the 

 recognition that the generation of noise by turbulence is equivalent to the action of 

 quadrupole acoustic sources. Then, when he came to write his paper on the scattering 

 of sound by turbulence, he worked out the analysis within the framework of the theory 

 of aerodynamic noise, and interpreted the scattered sound in terms of the action of 

 quadrupole sources. 



I avoided that presentation because I think it makes for unnecessary complica- 

 tion. In this problem of scattering, it is not really necessary to know that the turbulent 

 fluctuations in velocity are equivalent in their action to quadrupole sources of sound. 

 The multi-pole character of the equivalent sources does not play an important part 

 in scattering theory, essentially because the wavelength of the sound is here comparable 

 with the length-scale of the turbulence, whereas the sound generated by turbulence 

 has a much larger wave-length. 



I think that probably Professor Lighthill will agree with that. 



M. J. Lighthill 

 Yes. 



G. K. Batchelor 



The second reason applies mostly to Dr. Kraichnan's work. I felt that he was 



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