205 
APPENDIX II 46 
1. FUNDAMENTAL EQUATIONS 
and eliminate wu, between this equation and [5] 
r, 
p= 2 (p, + 4 pus — m) — 5 pu? + Po [8] 
The impulse at any point distant r from the center can be found from [6] or [7] 
fw — py) dt = prAu — x p[urdt = PA(pu, = 5 pfurde [9] 
where A denotes the change of a quantity during the time of integration. If r is 
large, fu 2dt can be neglected. 
APPENDIX II | 
2. OSCILLATIONS 
II. OSCILLATIONS OF A BUBBLE IN INCOMPRESSIBLE LIQUID 
Suppose now that the cavity contains gas of negligible mass; let the gas be- 
have adiabatically, losing or gaining energy only by doing work upon the liquid. An 
equation of motion for the bubble of gas can be obtained by putting in [7] p= Ps 
r= 7, andu = 4u,, carrying out the differentiation, and noting that wu, = dr, /dt: 
2 
R= P E (<2) + 1% <# + Dy [10] 
Multiplying this equation through by 7,2 dr,/dt and then integrating with respect to 
the time, we obtain successively 
dr, 3 dr,\>, 3 at% = dr, 
7-29 — —aegateal (ae at ey by eet 
Povo at | i ie ) + at dt2l* ™ ae 
2 
| pon? dr, = Pre ($2) + t Pot? + const. [11] 
Now the volume of the gas is 
4nrj 
vy = “3% [121 
hence 
2 See & odW: 
Baty Ory = Pat 4a [13] 
where Wis the energy of the gas. Thus we can write 
[roi dn, =— S + const. 
Then Equation [11] can be written 
dr,\2?_ C W 2 p 
SAE \ Ve eee Es ee) 1 
(a2) iia tenpre po <p ae 
where C is an arbitrary constant dependent on the initial conditions. If the gas is 
ideal,. and if the ratio y» of its specific heats is also constant, 
Wie fee, Pgvz = A= const. 
y-1 
and, using [12] we find : 
J heey (Sc Secale ana 
2npr, 4n 2mp(y—1) nr (15] 
