XVI 



WAVE SCATTERING DUE TO TURBULENCE 



G. K. Batchelor 

 Cavendish Laboratory, University of Cambridge 



Summary 



When a sound wave is incident upon a part of the medium which contains 

 small irregularities, scattered waves are set up. The irregularities, which may be either 

 variations of velocity or of physical properties such as temperature, are here assumed 

 to be random as a result of the action of turbulence in the medium. The basic 

 equations that describe the propagation of sound in both air and -water containing 

 various kinds of non-uniformity are derived. After a general exposition of the theory 

 of scattering in continuous media, the scattering cross-section for each of these cases 

 is obtained, and is found to depend on the spectrum of the spatial distribution of the 

 relevant property of the medium. 



It happens occasionally in science that a problem suddenly becomes important 

 and interesting in several different contexts simultaneously. This is exactly what has 

 happened in the case of propagation of waves through a medium whose physical prop- 

 erties are not quite uniform and vary from point to point in a random fashion as a 

 result of the presence of turbulence. This problem has sprung into prominence in the 

 post-war years in no fewer than four different physical settings. 



One setting concerns the propagation of radio waves of about one metre wave- 

 length through the atmosphere, and has led to the spectacular development of a new 

 method of long-range communication, which employs scattering of the radio waves 

 through a small angle by small random variations of electron density in the ionosphere. 

 Another concerns the old problem of the scintillation of starlight; that the twinkling of 

 stars is due to variations of refractive index in the atmosphere was known to Newton, 

 but only recently have attempts been made to analyse the phenomenon quantitatively 

 and to relate it to the turbulence in the atmosphere. In these two contexts the waves 

 are electro-magnetic, whereas in the other two I have in mind sound waves are 

 involved. 



Most people here will already know of the current interest in various aspects 

 of the propagation of sound in the ocean. If the ocean were perfectly uniform, ordi- 

 nary acoustic theory would tell us all that we need to know in order to design and 

 operate equipment for sound-ranging and sound-detection. But the ocean is not uniform, 

 and contains currents which convect the sound and variations of temperature and of 

 salt concentration which refract the sound. These non-uniformities are made random 

 by the ever-present turbulence, and are broken down in scale so that there are always 

 some Fourier components with wave-lengths comparable with that of the incident 

 sound. As a consequence, sound signals are subject to large fluctuations in amplitude, 

 to scattering about their original direction, and to rapid attenuation. This is not by any 

 means a new problem, but the needs of naval warfare gave a great stimulus to its 

 investigation, and interest in it has remained active during the post-war years. 



Finally, there is the allied problem of propagation of sound in the atmosphere. 



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