III-37 



This is the fundamental result for bubble-water mixtures with substantial 

 bubble-bubble interaction: a sound wave in such a medium propagates with an 

 effective velocity of propagation given by the real part of c, and an effective 



attenuation given by the imaginary part of k = - . 



c 



A few comments are in order about this result. First, the same re- 

 sult may be obtained by a somewhat more rigorous method which considers in 

 detail the combined effect of the many scattered wavelets. The method, due to 

 Foldy, is given in Appendix B. Second, several authors have obtained (III-35), 

 and all use the form of g(a) which corresponds to Models II and III. Meyer 

 and Skudrzyk derived a slightly different expression, which caused them to use 



the mechanical analog (Model I) form of g(a). In their paper they use — — as 



a 

 the total pressure on the air bubble, instead of Ap' . We see from (III-29) that 



/m \^ 



this would lead to a result which differs from ours by a factor ( — ) , and in- 

 deed they obtain c''' = c"^ + ^^§^^' instead of (III-35). However, since 







they choose Model I for g(a), they introduce a compensating factor of 



fej 



and their end result is therefore equivalent to that of other authors, provided 

 their damping constant is properly interpreted. 



Equation III-35 is easily modified to take into account a bubble popula- 

 tion which consists of different size bubbles. Suppose there are n(a)da bubbles 

 per unit volume each, with radius in the range between a and a + da. The con- 

 tribution of this portion of the bubble population to the change in total air volume 

 would be identical to that of (III- 34) but with n replaced by n(a)da. The total 

 change in air volume is then found by integrating over da, and this therefore 

 modifies the final result for c~^ to: 



+ ^ I da n(a) g(a) (III- 36) 



^ Tdanl 

 u^ J 



From (III- 35) or (III- 36) we can compute the relative sound speed in the bubbly 

 mixture as the real part of -^, and the attenuation of the intensity (which is 

 twice the attenuation of the pressure) as twice the imaginary part of - . In 

 particular, we may write (III- 35), using the Model III expression for g(c) as: 



;arthur a.HittUJnf. 



