392 



gas globe rises or sinks. Furthermore, effects due to compressibility of 

 the water, associated with the emission of acoustic radiation, have been 

 neglected. 



MIGRATION DUE TO GRAVITY ALONE 



If X is taken vertically upward. Equations [1] and [2] become, when 

 no boundary is near, 



_C^ ^ 2 r, . „2.r, 1 ;>2 , 2 



«^ = T7^ + ;|s /(P, - Po)R'dR -^X' + fg{X-X,) [22] 



v= 2£ 



pJR'dt [23] 



Here g is the acceleration due to gravity. 



Numerical integrations of these equations have been given by Taylor 

 (5) and others (6) (7) (8) The effect of the gas, as represented by the 

 occurrence of p^ in Equation [22], is usually not large; its smallness arises 

 from the fact that, in practical cases of motion due to gravity alone, the in- 

 ward motion of the water during each compression phase is arrested chiefly 

 not by the gas but as a consequence of the conversion of radial kinetic ener- 

 gy of the water into kinetic energy of translational motion. That is, when 

 the center of the gas globe is nearly stationary, the radial kinetic energy 

 of the inrushlng water becomes converted, at the instant of peak compression, 

 entirely into energy of compression of the gas; but if translational motion 

 of the gas globe occurs, part of the kinetic energy remains in the water in 

 association with the translational motion. For this reason the inward radial 

 motion is checked at a larger radius than when the globe Is stationary. In 

 migration due to gravity alone, nearly all of the energy usually thus remains 

 in the water, and the motion during the compression phase is nearly the same 

 as if no gas at all were present. Because of this conversion of the energy, 

 the radial oscillations gradually die out, as the velocity of rise increases, 

 especially if the hydrostatic pressure is very low or if the gas globe was 

 produced by a large charge. 



In UNDEX 10 (7), Figures 1 and 8, two plots are given, based upon 

 numerical integrations, from which estimates of the rise due to gravity can 

 be made for a wide range of charge weights and depths. These estimates agree 

 within 8 per cent with values calculated from the convenient approximate 

 formula 



H= 4-^-^ ■ [24] 



Pa 



