III-35 



The variations in the volume A V consist of two parts- -the change in vi^ater 

 volume AVyv and the change in the air volume A V^. Therefore, 



AV = AV + AV (III-31) 



w a 



Let us examine A Vg^ and see how it depends on Ap. Note that AV^ is just the 

 sum of the volume changes of each of the N = n Vq bubbles in V . Now, whereas 

 the macroscopic picture is that of a sound wave, Ap(x,t) = Ape^^"^ ^, propa- 

 gating through the mixed medium, the microscopic picture is quite different. As 

 a result of Ap, each bubble is caused to pulsate. Suppose the internal pressure 

 inside the bubble is caused to vary as Ap' (t) = Ap' e"^*^*^, and suppose further that 



ik„ r 



this gives rise to a scattered wave, Ap(t) = Ap e (where k^ = — = 



sc r Cq 



wave number in pure water and r is measured from the bubble center), in the 



Ap 

 sc 

 vicinity of each bubble . Near resonance, both Ap' and are much greater 



than A p. However, we know from Table III-l that Ap is proportional to Ap 

 according to: 



for models II, III 



(?)■ 



•1-16 



Ap = g(a)Ap where g(a) - / (III- 32) 



for model I 



¥1 



id 



We also note from (III- 29)that the internal pressure Ap' is proportional to the 



Ap 



SC 



scattered pressure , and in fact near resonance almost equal to it: 



Ap-.Ap il£2_ (i.ika)e^^"^''-^ (^Y(i-ika)e^^" (III-29a) 



^ ^sc ,.,a « „3 a I 111 I 



(i) 



Arthur B.ILittlcJnt. 



S-7001-0307 



