468 



ACOUSTIC THEORY OF BUBBLES 



On this picture the sound wave behaves as though 

 it were proceeding through a homogeneous medium, 

 in which the sound velocity is a smooth complex 

 function of position. 



Secondly, this picture must be supplemented to 

 take scattering into account. The sound waves sent 

 out from the different bubbles produce scattered 

 sound, which goes out in all directions. This scattered 

 radiation may be regarded as resulting from the fact 

 that in a random collection of point scatterers the 

 number of scatterings per unit volume is never 

 constant from one region to another, but shows 

 statistical fluctuations. A theory of the scattering 

 of light in air is given along these lines in a well- 

 known text on statistical mechanics.^ More simply, 

 the intensity of scattered radiation may be regarded 

 as proportional to the average squared pressure re- 

 sulting from all the individual bubbles. As the 

 bubbles move around, the relative phases of their 

 scattered wavelets will vary widely, and constructive 

 and destructive interference will be equally likely. 

 With this picture, the average squared pressure may 

 be regarded as simply the sum of the squares of the 

 pressures in each of the scattered wavelets. 



In ship wakes the number of bubbles in a small 

 volume is rarely sufficiently great to produce reflec- 

 tion and refraction of sound waves. The gradual at- 

 tenuation of the incident sound beam and the scat- 

 tering of sound energy in all directions by each 

 bubble individually are therefore the two effects of 

 greatest interest. 



The preceding discussion is, of course, not very 

 rigorous. The results stated here have been proved 

 rather generally, however, in an elegant solution to 

 the general problem.''^ This analysis makes certain 

 assumptions, the most important of which are that 

 the bubbles have diameters much smaller than the 

 wavelength of the incident sound, and that the 

 average distance between bubbles is much larger 

 than their dimensions. The solution, as a result of its 

 physical generality, is of considerable mathematical 

 complexity, and therefore will not be reproduced 

 here. But the mode of approach used in this general 

 theory will be briefly sketched. 



The chief feature of this theory is its use of con- 

 figurational averages. Different bubbles may be almost 

 any^vhere within a certain region. For each distribu- 

 tion of bubbles the sound pressure p at a given time 

 will have some definite value. If now an average value 

 of this pressure is taken for all possible positions of 

 the different bubbles, a configurational average of p, 



denoted by <p>, results. Thus is usually not equal 

 to the time average of p, since this time average 

 vanishes because of the oscillations of p between 

 positive and negative values. Similarly, <p-> may 

 be defined as the configurational average of p-. 



The simplified picture presented at the beginning 

 of this section may be given a precise meaning in 

 terms of these configurational averages. The quantity 

 <p> is found to obey the wave equation in a 

 homogeneous medium in which the complex sound 

 velocity is a function of position. This configurational 

 average acts in general as the pressure from a re- 

 fracted sound wave. Thus <p> gives rise to a trans- 

 mitted wave; after leaving the scattering region, this 

 transmitted wave bears a definite phase relationship 

 to the incident wave. 



For any particular configuration, the value of p 

 may differ from <p> . A measure of this difference is 

 provided by the mean square value of p — <p>, 

 which is equal to <p^> — <p>2. The analysis 

 shows that this difference is simply the sum of the 

 squares of the pressures in the sound wave sent out 

 from each of the bubbles. These additional terms 

 therefore represent just the scattered sound, includ- 

 ing sound that has been scattered several times. 

 Thus the intensity at any point, which is proportional 

 to p^, is on the average the sum of two terms; the 

 first term <p>^ represents the coherent wave, 

 propagating through a homogeneous medium in 

 which the sound velocity changes in some way with 

 changing position. The second term, <p-> — <p>- 

 represents the sum of the scattered waves from each 

 bubble. At any one time the value of p-, even when 

 averaged over a few cycles, will usually differ from 

 the sum of these two terms, but as the configuration 

 of bubbles changes, the time average of p- should ap- 

 proach the configurational average of p-. In most 

 practical situations a period of several seconds is 

 usually sufficient to bring the time average of p- 

 close to the configurational average. If, then, 

 averages are taken over time intervals of several 

 seconds, the simplified picture presented at the be- 

 ginning of the section may be taken as correct. 



When the average distance between the bubbles 

 becomes very small, or, in other words, as the 

 average number of bubbles per unit volume becomes 

 very large, this simplified picture becomes inadequate. 

 In this case, another ter^m must be included in < p'^ > , 

 in addition to the two terms representing the re- 

 fracted (coherent) wave and the scattered (incoher- 

 ent) waves. This term is difficult to interpret, but 



