MOTION OF THE GAS SPHERE 305 



films showing the migration of the bubble toward model rigid surfaces 

 and targets. Similar small scale records of such motions, which are of 

 considerable importance in assessing the role of secondary bubble pres- 

 sures in explosion damage, have been taken by Campbell (17) at the 

 Taylor Model Basin, using detonators, but without the reduction of 

 surface pressure necessary to give a model scale of large explosions. 



Campbell and Wyckoff (18) have made streak silhouette and strobo- 

 scopic pictures, using continuously running film, of half cap and Num- 

 ber 6 detonator cap explosions under reduced atmospheric pressures in 

 a 24 inch water tunnel. The records obtained illustrate very beauti- 

 fully the changes in maximum size and period with hydrostatic pres- 

 sure, reveal interesting cavitation regions at various times between the 

 shock wave front and the bubble surface, and also show the relation of 

 bubble motion to displacements of water at the free surface. The 

 energy values for these very small explosions are not too precisely known 

 and detailed analysis of the radius and period data is not attempted 

 here. The other observed phenomena are considered in Chapter 10. 



8.7. Effects of Compressibility and Nonspherical Form on 

 Bubble Motion 



We have so far developed the approximate theories of bubble pul- 

 sation and migration on the simplifying assumptions that the bubble 

 retained a spherical form and that the surrounding water was incom- 

 pressible. Although most of the calculations of the motion suitable for 

 comparison with experimental data have been made with these approx- 

 imations, more exact formulations undertaken by Herring and others 

 are of interest in indicating the nature of the errors to be expected from 

 the simpler calculations. There are described here Herring's develop- 

 ment (46) of the equations of motion for spherical symmetry, including 

 the effect of compressibility and of gravity on the motion, and the 

 analysis by Penney and Price (86) of the stability of the spherical form 

 during the pulsation. 



A. The effect of compressibility. The equation of an inviscid fluid 

 for spherical symmetry is, from section 2.3, 



du ^ _ _ 1^ 



dt dr p dr 



where u is the radial velocity and P the pressure at a point (r, t) . Inte- 

 grating this equation from the bubble surface (r = a) to infinity gives 



«""S+l(l)'-"f'<">+/>S'"=-J 



dP 

 P 



