207 
APPENDIX II | 48 
2. OSCILLATIONS 
negligible over most of the range from 7, to rz and especially near 1, = rz. Let us, 
therefore, drop this term altogether. Then [16] gives 
Zaps 
== ° 20 
C Taras [20] 
Furthermore, since the range of integration greatly exceeds r,, little error will re- 
sult if we also extend the range back to r, = 0. Then [17], [14] and [20] give 
a r, 
2 
reer eae 
0 
Write 
—— r, sin’ 6 , dr, = (=) rsin *6 cos eae 
Then 
The integral can be expressed in terms of gamma functions or evaluated numerically; 
its value is 1.124. 
In the applications, however, it is more convenient to express JT in terms of 
the maximum energy W of the gas, which is its energy at minimum size, when 1, = 1. 
Since r2/r, is assumed large, it is evident from [20] that, when T, = 7%, the first 
term in [16] is much larger than the third and must, therefore, be nearly equal to the 
second. Hence, with the help of [20], 
W, 27-8 me ei Le ([21a,b] 
2ap Oe Sphere Ge = 
approximately. Thus, evaluating the constants, 
JP SS W533 eR P Sel Apa Dy fi J 
valid for large rs/r, (perhaps re/7, >10). 
APPENDIX II | 
3. PRESSURE AND IMPULSE 
III. PRESSURE AND IMPULSE IN THE LIQUID 
The pressure p at any point in the liquid, as the bubble oscillates, is 
given by [8] in terms of the pressure p, of the gas and the velocity u, or dr,/dt of 
the interface, which is given in turn by [14]. Or, in Equation [7] the pressure is 
given in terms of r, and u,. To find p as a function of the time, Equation [14] must 
be integrated. 
Expressions for the impulse in the liquid are easily obtained from [6] and 
