49 



P A _ «L\ + 3 P R_ = o 



1 Ro 3 ^ C Rn 



where P is the initial pressure in the cavity. Since the actual conditions 

 within the cavity are not accurately represented by the Rayleigh problem, this, 

 will not be discussed further here. The results are, of course, of some use 

 in estimating pressures for cavities which are completely air-filled, and do 

 show that very high pressures may arise. Osborne, 46 assuming adiabatic com- 

 pression, obtained results which agreed well with his experiments in which the 

 cavities contained large quantities of air. These solutions neglect surface 

 tension, however, and the possibilities of retardation of the motion by the 

 finite rate of vaporization and condensation in an actual cavity. Neverthe- 

 less, the wall velocity given by these results are fairly good approximations 

 in the beginning of the motion, as shown by the experimental results of 

 Mueller 45 and of Knapp and Hollander. 47 



A similar computation of bubble collapse was carried out by Ackeret 10 

 for densely packed, spherical bubbles, assuming a polytropic law of compres- 

 sion. However, the same objections apply here that are mentioned above. 

 Since these theories lead to pressures which are obviously too high, both 

 Ackeret 10 and Knapp and Hollander, 47 as well as a number of other writers re- 

 porting between the appearance of these two references, propose that the pres- 

 sures at collapse may be estimated by the usual theory of water-hammer, provid- 

 ing the impact velocity is known, i.e., P = pcU where c is the velocity of 

 sound in the liquid. Even if the velocity could be obtained accurately for 

 cavities which contain no permanent gas, the result would not correspond with 

 most cases since entrained air and air diffused during several cycles would 

 result in smaller pressures than predicted by such a theory. 



Another analysis was carried out by Silver. 75 In Silver's computa- 

 tion, the rate of collapse is assumed to be governed solely by the rate at 

 which the vapor in the cavity can be condensed. Thus, although there is the 

 serious objection that inertial effects are ignored, this computation might be 

 taken to represent a lower limit for a cavity that is completely vapor filled — 

 ths Rayleigh (empty) cavity representing an upper limit. For this reason, 

 the results are given here. (Although Plesset recognized that the rate of 

 condensation might become of importance in the final stages of the collapse, 

 it was not important over the range of his numerical computations, and he did 

 not include it.) Assuming that the rate of condensation is proportional to 

 the rate of thermal conduction away from the surface of the cavity, and that 

 the rate of thermal conduction is proportional to the difference between the 

 saturation temperature T at vapor pressure p and the ambient temperature T Q , 

 Silver related the rate of bubble collapse to the pressure difference using 



