Van Wijngaarden 



the collapse of cavitation bubbles, filled with vapor and a small amount of air, 

 are in fact the cause of cavitation damage. For a bubble with initial pressure 

 Pq and collapsing under an outside pressure Pco the rate of change of the radius 

 R with time t , neglecting vapor content, will be seen from the next section to be 

 given by 



The minimum radius appears to be about SpoRg/p^ and the associated pressure 

 in the bubble can raise considerably above p^. The question arises how in a 

 mixture of collapsing bubbles the average pressure in the fluid behaves. This 

 question was considered in Van Wijngaarden (2) and is discussed here again in 

 connection with the results of linear theory. 



The investigation presented here attempts to link single bubble theory with 

 homogeneous fluid theory both for linear and nonlinear oscillations of the indi- 

 vidual bubbles in the mixture. 



2. LINEAR THEORY I 



To fix ideas we consider the two-dimensional problem of a bubble-water 

 mixture bounded at y' = by a solid wall and at y' = h by pure water. At times 

 t' < the pressure both in the pure water and in the bubble-water mixture is p^ 



At t' = 0* the pressure in the region y' >h is increased by an amount Ap. 

 This gives in the mixture rise to variation of average pressure p, and average 

 velocity v . The averaging is over a region small with respect to h but still 

 containing many bubbles, so that it is required that 



,-1/3 



« h . (2.1) 



The time history of the radius of an individual bubble, neglecting the effects of 

 viscosity and surface tension, is given by (see e.g., Lamb (3)) 



R ^'^ 



dt' 



|©l-.-p. (") 



in which Pg is the pressure of the gas in the bubble. Equation (2.2) has been 

 used by many authors in order to deal with the collapse of a single bubble under 

 influence of pressure p^, far from the bubble. 



Writing p = p<„ and assuming a polytropic change of the bubble pressure with 

 bubble volume given by 



PgR^=Pg^RoV (2.3) 



the relation (1.3) can be easily derived from (2.2). 



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