III-18 



Thus, we have one relation between the three amplitudes Pinc Psc ^"*^ P' • ^ 

 second relation may be obtained from the equation of motion of the water just 

 outside the bubble boundary. Consider (III-2a) in radial coordinates, with Uj. 

 as the radial component of the velocity: 



i UJ Po Uj. = p ^r (III-25) 



Since the incident wave does not depend on r, the gradient of the pressure on the 

 bubble surface may be obtained entirely from the scattered wave: 



ika-ia)t 



p (r=a) = p (r=a) = p (ik--)- (III-26) 



^ ,r sc,r sc a a 



It remains to evaluate the radial velocity Uj. in terms of p' by analyzing the thermo- 

 dynamics of the interior of the bubble. For a perfect gas undergoing adiabatic com- 

 pression, the volume V is related to the total pressure p = po + p' according to 

 p ~ V" ^ where Y is approximately 1.4. The volume for a bubble with radius 



da 

 a + da is given by V = 4/3 rr (a-t-da)^ "^ Vq (1 + 3 — ) where Vq is the unper- 



4 3 ^ 



turbed bubble volume -5- ""^ a . It follows that pressure changes inside the bubble 

 are related to changes of bubble radius according to: 



^ = _ I^ . - 3 Y - (III-27) 



p V a 



The acoustic fluctuations dp = p' and da are very small compared to the stagna- 



da 

 tion values p and a. Since — is just the radial velocity u^. of the bubble surface, 



we can rewrite (III-27) to first order in the acoustic quantities as: 



J_ dp; ^ . 3y ^ = - i^ p- (III-28) 



Pn dt a r p^ 



Arthur B.IlittkJnt. 



S-7001-0307 



