272 MOTION OF THE GAS SPHERE 



(the secondary or bubble pulses) are emitted, which an incompressive 

 theory can describe only as a disturbance appearing simultaneously at 

 all points in the liquid. 



From the foregoing considerations, it is to be expected that the 

 theories based on noncompressive flow will be most successful in ac- 

 counting for the general properties of the motion in its expanded phases, 

 and in predicting properties, such as the period, primarily thus deter- 

 mined. The vertical displacement of the bubble at its minimum and 

 the characteristics of the pressure wave emitted at that time are, how- 

 ever, more seriously affected by the approximations. In addition to 

 the common assumptions just described all the theories involve other 

 approximations, and the degree of refinement of the various treatments 

 is largely a question of the extent to which these further approximations 

 and idealizations are removed. 



8.2. NoNCOMPKESsivE Radial Motion Neglecting Gravity 



The simplest approximation to the true motion of the gas bubble is 

 the one in which it is assumed that the motion of the surrounding water 

 is entirely radial and there is no vertical migration. In this approxi- 

 mation, which has been discu sed by a number of writers, the hydro- 

 static buoyancy resulting from differences in hydrostatic pressure at 

 different depths is neglected. It is thus assumed that at an infinite 

 distance from the bubble in any direction the pressure has the same 

 value as the initial hydrostatic pressure Po at the depth of the charge 

 (atmospheric plus the added pressure of the water column). For a 

 given depth of charge, the differences in pressure at the surface or near 

 the bubble will clearly be greater the larger the charge and bubble re- 

 sulting from its deformation. The neglect of differences in hydrostatic 

 pressure should thus be more serious for large charges and small depths. 



If radial flow is assumed, the equations of continuity and motion for 

 the water are (Eqs. (2.2) and (2.4)) 



(8.1) ^ + u^' + p^ + ^-^ = 



dt dr dr r 



du ^ du , dP ^ 

 dt dr dr 



For pressure changes of the order 15 Ib./in.^, such as prevail over most 

 of the bubble motion, the corresponding changes in density are of the 

 order 10~^ po, where po is the equilibrium density. Under these condi- 

 tions, the derivatives of density p are easily seen to be negligible in the 

 first of Eqs. (8.1), which then becomes 



