The only difference between the scattering cross-sections for stationary air and sea- 

 water, apart from the difference in the numerical factors, lies in the appearance of the 

 cos 2 # factor in the former but not in the latter. In the case of sea-water, the scat- 

 tered energy flux is fairly evenly distributed over all directions, but in the case of air 



there is a minimum at the lateral plane 9 = — . The expressions for the total energy 



flux Ja(l)dl in the cases of air and water should not differ by more than a factor of 

 order unity as a result of the presence or absence of the cos 2 #. Much more substantial 

 differences can occur as a result of differences between the forms, and the magnitudes, 

 of the spectra of density fluctuations in the one case and temperature fluctuations in the 

 other. However, these differences cannot be described in a general way since they 

 depend on particular considerations such as the value of the sound frequency. 



Uniform medium in turbulent motion 



A discussion of the effect of turbulent fluctuations of velocity on the propaga- 

 tion of sound waves encounters many difficulties, and a good deal of care is necessary 

 in the use of approximations. Many papers have been written on the problem, and 

 three people have independently derived the cross-section for single scattering correctly. 

 The first was Blokhintzev in 1945, who made use of the equation for the propagation 

 of sound in turbulent flow derived by Obukhoff two years earlier, and the others were 

 Lighthill and Kraichnan in 1953. These people used completely different kinds of 

 argument, and it takes a good deal of work to show that in fact they used the same 

 assumptions and arrived at the same results. None of these pieces of work seems yet 

 to have been assimilated into the literature of the subject. By standing on the shoulders 

 of these pioneers I have devised my own derivation, which I believe to be simpler and 

 freer from logical imperfections. The result is important enough to warrant continued 

 attempts to improve the argument. 



One or two numbers will help us first to see what is perturbing what, in this 

 combination of sound and turbulence. The r.m.s. particle velocity in a plane sound 

 wave of N decibels sound level in air is about 5 x l(h 6 x 10 N / 20 cm/sec, which comes 

 to 0.5 cm/sec for a very loud noise of 100 decibels. On the other hand, the r.m.s. 

 of the turbulent velocity fluctuations in the lower levels of the atmosphere is often in 

 the neighbourhood of 50 cm/sec. Thus the fluid velocity in the turbulence field alone 

 is very much bigger than that in the sound field alone. However, this is not neces- 

 sarily true also of the corresponding fluctuations in density and pressure, since density 

 variations in a sound wave increase as the first power of the accompanying particle 

 velocity whereas in a turbulent motion they increase as the second power. Use of 

 the above figures for the atmosphere shows that the density fluctuations in the sound 

 and turbulence fields alone are comparable in magnitude. Similar calculations for 

 underwater sound waves, having intensities like those commonly used in connection 

 with ranging equipment, show that here again the sound waves are likely to be 

 accompanied by particle velocities much smaller than, and density fluctuations com- 

 parable to, those produced by turbulence in the sea. 



Despite this approximate parity of the separate density variations, the incident 

 sound wave cannot have much effect on the turbulent velocity field. The shearing 

 motions of the turbulence are determined primarily by the boundary conditions on 

 velocity, and are not affected appreciably by superimposed relative density variations of 

 order M t 2 (where M t is the Mach number of the turbulence alone) — as indeed can 

 be seen from the fact that the velocity distribution in the pure turbulence is known 

 to be approximately independent of M t . Thus the velocity distribution in the combined 

 incident sound and turbulence field is approximately the same as that in the turbulence 

 alone. 



419 



