27 



(A4>) 



M -1 i 

 at r [ 



R*R< 



2RR + r*r; 



Evaluating the last equation at r = R, 



(4tf) 2 = R 2 



Mth these results, Plesset obtains the equation of motion of the surface of 

 the cavity as follows: 



1 [p(R) - P(t)] =-|R ; 



+ RR 



Rayleigh's case is obtained by putting P(t) - p(R) = P = constant. Plesset 

 takes p(R) = p — ^-, and thus includes the effects of surface tension. Tak- 

 ing as boundary conditions, R = R max at R = 0, Plesset integrated the equatio: 

 of motion numerically, for the pressure field corresponding to the conditions 

 of the experiments of Knapp and Hollander. The results of the analysis for 

 one cavity are shown in Figure 17 which is taken from Reference 51. It is 

 understood that Plesset has attributed the discrepancies on the right side of 

 Figure 17 to wall proximity and has successfully accounted for them on this 

 basis. 



It is clear that analyses 

 based on empty cavities in an incom- 

 pressible liquid cannot give rise to 

 oscillations. Furthermore, with con- 

 stant external pressure, the velocity 

 R increases without limit as the bub- 

 ble collapses. To obviate this result. 

 Rayleigh extended his computations to 

 include the case of a cavity filled 

 with a gas which is expanded and com- 

 pressed isothermally and showed that 

 the boundary oscillates between two 

 positions, of which one is the initial 

 position. Although the motion of the 

 oscillating cavity in a real liquid is 

 evidently complicated by the diffusion 



Figure 17 - Comparison of Theoretical 

 and Measured Cavity Radii from 

 Plesset ' s Analysis— Reproduced 

 from Reference 51 



