The effect of non-uniformities in the atmosphere came under notice during the first 

 world war in connection with devices for the detection of enemy aircraft, but these 

 devices were not successful and the problem was not examined fully. Interest in it has 

 more recently been revived, as a part of a comprehensive investigation of the noise 

 produced by jet engines. Instead of wanting to pick up the noise of an aircraft, we 

 now wish to suppress or avoid it, but in either case we require a knowledge of the 

 effect of random variations of the properties of the air on the sound waves passing 

 through it. 



Thus radio engineers, astronomers, oceanographers and aerodynamicists all 

 have good reasons for wanting to know more about the same basic problem. So far 

 as I can tell from a limited acquaintance with the separate literatures, they seem on 

 the whole to have pursued the problem in their own ways, without making much use 

 of advances made in other physical contexts. There is a need, I think, for an investiga- 

 tion which is framed in as general a manner as possible, and which brings together 

 all the available results. The general aspects of the subject are especially suitable as 

 material for a survey at the present time, and, although this lecture does not provide 

 the time that would be needed for a proper account, I hope at least to arouse your 

 interest and to give you the right orientation. 



In order to keep fairly close to the theme of this symposium, I propose to 

 confine my attention to sound waves, thereby excluding the applications of the theory 

 to radio communication and stellar scintillation. (The analysis required in these latter 

 applications is essentially similar, such differences as there are being mostly those 

 arising from the need to specify electromagnetic waves by a vector, instead of a scalar, 

 quantity.) 



When reading the published papers on the propagation of sound waves in a 

 non-uniform medium, I have noticed that many of the authors assume intuitively the 

 validity of a certain equation describing the propagation of sound, and then proceed 

 to investigate some special feature of this equation, such as the nature of the scatter- 

 ing. This procedure has its dangers, since mistakes in the form of the governing 

 equations cannot be detected by the usual checks on consistency of the analysis. Some 

 authors have assumed, for instance, that any non-uniformity of the physical properties 

 of the medium is equivalent to a variation of the phase velocity, that is, of refractive 

 index; others have assumed that the effect of turbulent convection with velocity u in 

 the medium can be represented by replacing d/dt in the conventional acoustic wave 

 equation by d/dt + u • V . Both of these ideas are plausible, but actually they are 

 too simple, and lead to correct results only in special circumstances. In view of the 

 evident differences of Opinion about the equations on which the whole problem is 

 based, I shall spend a large part of my time on a careful derivation of the governing 

 equations appropriate to each kind of non-uniformity of the medium. I regret that 

 there will not be time to get to grips with the observational data about scattering of 

 sound. 



There are three essentially different aspects of a theoretical treatment of the 

 problem of propagation of sound waves through a non-uniform medium in turbulent 

 motion: 



a. the derivation of the equations describing the propagation of sound waves in 

 these circumstances; 



b. the analysis of the scattering and of other consequences of the non-uniformity; 



c. the determination, from turbulence theory or experiment, of those statistical 

 properties of the medium that the analysis in b. has shown to be relevant. 



Consideration of this last aspect would take us into the complexities of turbulence 

 theory, and must be omitted for lack of time. I shall begin with an analysis of scatter- 

 ing in general form, since talk about the form of governing equations is not interesting 

 until one knows the kind of use to which they are to be put. 



410 



