466 



ACOUSTIC THEORY OF BUBBLES 



By combining this expression with equations (3) and 

 (32), the cross section for extinction is finally ob- 

 tained: 





(43) 



+ 52 



The extinguished energy is obviously the sum of 

 the scattered and absorbed energy. Therefore, the 

 absorption cross section Ca of the actual bubble can 

 be defined by the relation 



Te = a, + Ca (44) 



and is thus found to be, from equations (34) and (43), 



AttR- 



e-o 



Oa^ 



iM 



(45) 



+ 52 



Note also the simple relation 



Os 



(46) 



A word must be said now about the function 5, de- 

 fined in equation (32). If fi and Ci are put equal to 

 zero, for the case of an ideal bubble, 6 reduces to )?, 

 and it is seen that equation (22) is indeed the correct 

 limiting form of equation (34). Numerical values of 

 /3 and Ci can be derived by an analysis of the several 

 physical processes known to contribute to the absorp- 

 tion of sound by the bubble — for instance, heat con- 

 duction, viscosity, surface tension, and other proc- 

 esses. There are also methods for determining 5 

 empirically from certain observations which will be 

 discussed. Inspection of Figure 2, which shows the 

 damping constant at resonance as a function of fre- 

 quency, will reveal that the predicted values of 5 are 

 much smaller than the observed ones. This discrep- 

 ancy indicates that some relevant physical processes 

 must have been overlooked in the theoretical anal- 

 ysis of the absorption effects. Hence, theoretical 

 evaluation of |S and Ci, although carried out else- 

 where,^ will be omitted from this review, and the 

 empirical values of 6 will be used for the interpreta- 

 tion of the acoustic properties of wakes to be given 

 in Chapter 34. 



The physical significance of 5 can best be visualized 

 by plotting c^/'^'kR^ against ///,. A resonance curve, 

 similar to Figure 1, is thus obtained. The peak value 

 of this graph is, according to equation (34), 



0.4 



0.3 



S, 0.2 



0.1 



10 15 20 25 30 35 

 FREQUENCY IN KG 



40 



Values found from oscillation of a single bubble 

 Values found from transmission through bubble screen 

 Adopted values of 5r 

 Theoretical curve for air bubbles 

 Theoretical curve for ideal bubbles 



Figure 2. Damping constant at resonance. 



^tW- 



fr 



(47) 



where 6r is the resonance value of h shown in Figure 

 2. If (rs(/) denotes the cross section for any non- 

 resonant frequency, it follows from equations (40) 

 and (47) that 



<ra(/) _ 



t 



(f-O- 



(48) 



+ 5H/,fl) 



Over a narrow range of frequencies near the peak of 

 the resonance curve, S{f,R) in the denominator of 

 equation (48) may be replaced by its resonance value 

 6r, and fr/f is very close to one. Hence, using the 

 abbreviation q = fr/f— 1, 



ih^h 



q'iq + 2y = 4q^ 



and equation (48) becomes approximately 



<r.(/) 1 



Csr 



4(?2' 



(49) 



(50) 



in other words, for any given small departure q from 

 the resonance frequency, the decline of as from its 

 peak value is sharper for greater values of l/6r or for 

 smaller values of 5r itself. The greater the sharpness of 



