M. J. Lighthill 



I hadn't actually prepared anything, but I had a feeling Dr. Batchelor would 

 say some interesting things, and I was right. 



There is this question about any differences between the approach he has 

 described and the approaches that Dr. Kraichnan and myself derived in 1953. I found 

 it a little difficult to see just what the differences were, in the mathematical details 

 of what he showed us and the thing that I did. 



But I would like just to mention something which isn't connected with mathe- 

 matical details, but rather with a physical interpretation which I found helped me, 

 though it won't necessarily help everyone. That was this. 



When Lord Rayleigh worked out the explanation of the blue colour of the 

 sky, he showed that this was due to variations of refractive index. Later, the theory 

 of refractive index was worked out, and it was seen that this was due to polarization 

 of dipole molecules in the direction of the incident beam. So we can say that this 

 scattering of light (with some preference for the shorter waves) is due to polarization 

 of molecules in a random sort of way, with some sort of spatial correlation. 



The point of view from which I found it helpful to look at the problem we have 

 just heard about was this. We have the following equation which Dr. Batchelor used — 

 I deliberately leave out any question of variation of properties and just leave in the 

 terms 



d 2 p d 2 



- a 2 V 2 p = (pUiUj). (1) 



dt 2 dxidxj 



We have this equation, and this equation also came out in the theory of aerodynamic 

 noise, and one said that the sound produced by turbulence was effectively that due to 

 the distribution of quadrupoles indicated on the right of (1), namely quadrupoles of 

 strength 



pUiUj (2) 



per unit volume. 



Now, Dr. Batchelor pointed out that in the presence of an incident wave the 

 term inside the second derivative in ( 1 ) would become 



p{Ui + Vi)(Uj + Vj), (3) 



where v t is the velocity field of the incident sound wave. We can write (3), the new 

 quadrupole strength per unit volume, as 



p{uiUj + UiVj + ViUj + ihVj). (4) 



Now, here you have got four lots of quadrupoles, and we can interpret them 

 physically as follows. The first term (see (2)) represents the sound that is being 

 generated by the turbulence itself, which we are not concerned with. The middle two 

 terms represent the scattered sound due to the interaction of the turbulence and the 

 sound beam. And the last term pv;v ? - represents the effect of the sound wave on itself, 

 which we know is a steepening, tending to produce shock waves, and so on, but will be 

 very small in the case of ordinary weak waves. 



The middle two terms 



p(UiVj + ViUj) (5) 



are the part we are interested in, and it struck me that a way of looking at this part 

 would be to say that here you had quadrupoles that were polarized in the direction of 

 the incident sound beam v,, you see. Well, only one axis of each quadrupole was 

 polarized — in the first term of (5) it is the v ; , in the second the v,. But you have an 

 analogy in this to the case of light scattering. 



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