where the spring constant, k , is given by 



; f f"?te 



2 4 

 Periodic motion occurs for k > 4v /R , no oscillations take place 

 2 4 so' 2 4 



when k = 4v /R , and exponential growth occurs for k < 4v /R . The 

 s o' r so 



"damped" case is not really one of great interest for we find that with 

 water at one atmosphere of pressure difference the cut-off radius below 



_Q 



which the bubbles do not oscillate is about 1.4 x 10 in. This is such a 

 small number that the viscous effect may be neglected for our purposes. 

 Likewise the compressibility damping effect may also be neglected (Clay 

 and Medwin 1977, Chapter 6). The natural frequency then is closely 

 equal to 



n = I -^ p~-p„ + ^r- > (2D 



on eliminating k and K. We see immediately that this frequency vanishes 

 when 



4a 

 Vv = " 3iT (22 > 



These conditions may be termed the critical ones and the value of R , so 



J o 



found for a given gas content of bubble (or value of K) the critical radius, 



R . . Figure 34 shows plots of these equilibrium radii and the locus of 

 crit & f h 



points (Equation (22)) separating stable and unstable regions. Some of the 

 numbers from Equations (21) and (22) are interesting, and for reference we 

 summarize a few in Table 2. 



61 



