290 MOTION OF THE GAS SPHERE 



fore two unknowns, U and a. To determine these a second equation is 

 necessary. 



The second relation is simply obtained from momentum consider- 

 ations. From the derivation for kinetic energy of the moving water, it 

 was found that its effective inertia as regards translation was that of a 

 mass 2irpoa^/3 with velocity U. The vertical momentum acquired by 

 the water should, therefore, be {2irpoa^/S)U and is the result of the 

 buoyant force on the gas sphere, equal to iirpoa^g/S by Archimedes' 

 principle. Equating the impulse of this force to the momentum ac- 

 quired, by Newton's second law, gives 



(8.24) j Fdt = j p„g J 



ahlt = — Poa^U 



and hence 



(8.25) U = -^ = ^ \ aHt 



at 



-I/- 



a result originally given by Herring. 



This rather intuitive argument may perhaps not be wholly satisfy- 

 ing to some. It should be pointed out that its validity is insured only 

 if artificial external constraints on the boundary, which must strictly be 

 introduced to keep it spherical, are such that they exert no resultant 

 force on the boundary and hence do no work on the bubble during its 

 motion. An equivalent condition is the requirement that the resultant 

 effect of the fluid pressure at the boundary on its motion be zero, and if 

 this condition is worked out, one obtains Eq. (8.24). To do this, one 

 must first compute the pressure Pa at the boundary surface r = a from 

 Bernoulh's equation (Eq. (8.18)) and the velocity potential. This pres- 

 sure acts normal to the surface and the resultant force in the direction 

 of motion {6 = 0) is obtained by integrating Pa cos 6 over the surface. 

 Setting the result equal to zero gives Eq. (8.24). 



B. Taylor^ s nondimensional form of the equations. It is readily seen 

 that the equations of motion, (8.23) and (8.25), do not permit simple 

 geometrical scaling, for which the equations are unchanged if all lengths 

 and times are multiplied by the same factor. This principle of similar- 

 ity, valid for shock and detonation waves, cannot therefore be applied 

 to bubble phenomena. The reason for the failure of this scaling is of 

 course the effect of gravity in addition to the internal gas pressure in 

 determining the motion. The initial energy of the gas is, however, 

 relatively small if one excludes a fairly small portion of the oscillation 

 cycle when the bubble is near its minimum size. Over most of the cycle 

 it is therefore reasonable, as an approximation in determining the gen- 



