212 



APPENDIX 



MATHEMATICAL THEORY OF RADIAL MOTION 

 AROUND GAS SPHERES IN WATER 



Consider a sphere of gas behaving adlabatlcally, so that, if p and 

 V are its pressure and volume respectively, or R its radius, and if Vp and iJj 

 denote values under the hydrostatic pressure p^, then 



pX = PoK 



whence 



P,= Po(ff' [26] 



Here y is the ratio of the specific heat of the gas at constant pressure to 

 its specific heat at constant volume. The energy of such a gas is 



or, if y = V5, 



PV _ 47r ^s/Rof'-' 



-1 3(y-l) 



-r) [271 



R * 

 W - 4np,-^ [27a] 



THE RADIAL MOTION 



By inserting the value of H^ f rora Equation [27] and also Vg = R in 

 Equation [lU] on page 46 of TMB Report 480 (4), the fundamental equation of 

 radial motion for the sphere of gas in incompressible water under steady 

 hydrostatic pressure p^ is obtained in the form 



(dR^' ^ Ci^ _ ^ 1 Po (Ro)'' _ 1 Po r281 

 ^dt ' R^ Z y - \ p ^ R' 3p 



where p denotes the density of water and Cj a constant whose value depends 



upon initial conditions. 



The maximum and minimum radii of the oscillating sphere, R^ and R^ 



respectively, occur when dR/dt = and hence are those values of R which make 



the right-hand member of Equation [28] vanish. Hence R^ and R^ are connected 



with each other and with Cj as follows: 



