206 
47 | APPENDIX II 
2. OSCILLATIONS 
In any case, let the constant of integration in W be so chosen that W>0. 
Since W>~oas r,>0, the right-hand member of [14] is negative at r, = O and 
at r,=0. If C is large enough, however, it. will be positive between two values 
T, = 7, and r, = r2 which are the roots of 
G Ww 2 
= -—-=f=0 [16] 
' ip 27p i 3p 
Then the bubble will oscillate between the radii 7, and rz with a period T given by 
_ of 2dr) 
T= 2) faa ote 17] 
1 
the integrand being given in terms of 7, by [14]. Even if the gas is ideal, the inte- 
gration involves, in general, unfamiliar functions. 
1. SMALL OSCILLATIONS 
Let r, be the value of 7, at which the bubble is in equilibrium, with its 
pressure Pp, = P+ If it is slightly disturbed from this position, it will execute 
simple harmonic oscillations about 7% = 7). For such motions, we can neglect (dr,/dt)? 
in Equation [10] and also put r, = 7) obtaining 
dr, 1 ( 
= rao Par By) 
dt” PT, 
Since p, - p, is small, we can write 
Py- Py = (1% — 7) —Po 
The period of the oscillation is, therefore, from the usual formula for 
harmonic oscillations, 
d pd 
r= an org (- $2) i: [18] 
the derivative being evaluated at 7 = 1%. 
If the gas is ideal, so that pv,” = p, (4r73/3)” = const., 
Hence 
T, = ann Va > [19] 
0 
2. LARGE OSCILLATIONS 
As the amplitude of the oscillation increases, the period, given by [17], 
increases slowly. For r,/r, = 5.9, numerical integration gives T = 1.30 T, (20). 
When the ratio r,/r, becomes lerge, e good approximation can be obtained as follows: 
At r= 7, the term containing Win [16] or on the right in [14] cannot ex- 
ceed the term containing C, else the whole would be negative. As r increases from 7), 
the Wterm decreases faster than the Cterm. Hence, if r2/7, is large, the W term is 
