12 



being carried out to several terms in an expansion of the velocity potential in Legendre poly- 

 nomials. However, Rattray showed that for the above bubbles, the time of collapse could hS 

 increased by as much as 20 percent for the assumed distance of approximately one-half to 

 one diameter from the wall. 



It is clear that analyses based on empty cavities in an incompressible liquid cannot 

 give rise to oscillations. Furthermore, with constant external pressure, the velocity of the 

 bubble wall increases without limit as the bubble collapses. To obviate this result, Lord 

 Rayleigh, who gave the first complete solution of the collapse of an empty, spherical cavity 

 in an incompressible liquid ^^ extended his computations to include the case of a cavity filled 

 with a gas which is expanded and compressed 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 and vapor- 

 ization processes and the problems of energy dissipation, such solutions, which are clearly 

 oversimplifications, nevertheless, give a clear, quantitative picture of the hydrodynamics of 

 the motion as long as the cavity radius is large compared with the minimum radius. 



In Reference 3, the writer pointed out that further extensions to include the problems 

 of energy dissipation associated with the compressibility of the vapor and gas mixture and of 

 the liquid would be of great practical as well as theoretical interest. For a bubble filled with 

 a permanent gas being compressed and expanded adiabatically rather than isothermally, Ray- 

 leigh's case corresponds to the case of pulsation of a gas globe following an underwater ex- 

 plosion (see, e.g.. Reference 19). Although, in the latter problems, much progress had been 

 made (up to the time Reference 3 was written) in describing an oscillatory motion with energy 

 dissipation, the problem of the vapor condensation and formation prevents a complete analogy 

 to gas-globe theory and a completely satisfactory description of the motion of cavities based 

 on this theory had not been formulated. However, if one neglects the condensation problems, 

 which are important only insofar as determining the conditions under which the vapor begins 

 to act as a permanent gas, many of the results and methods first developed in the field of ex- 

 plosion hydrodynamics may be applied to the present problem. 



This has been done within the last two years, first by Trilling^° and later by Gilmore. -^^ 

 Trilling derived the velocity and pressure fields about a bubble in a slightly compressible 

 liquid using the acoustic approximation (i.e., only velocities that are first order small com- 

 pared with the velocity of sound are considered) and obtained results which coincides with 

 those of C. Herring -^-^'^^ who carried out the same computation some years earlier, but only 

 for the conditions at the bubble surface. Since these results are included in the later exten- 

 sion of Gilmore,-^^ only one result of interest will be mentioned in connection with Trilling's 

 work. Computations of the collapse of a bubble supposed to be filled with a perfect gas 

 showed that a series of shock waves are propagated into the gas. However, it was shown that 

 the average variation at the bubble wall is very nearly the same as if the gas were compressed 

 uniformly and isentropically. 



