332 MOTION OF THE GAS SPHERE 



flow must be such as to bring the surface nearer to the wall. A con- 

 siderable amount of momentum is imparted to a large mass of water in 

 this way when the bubble is large. As the bubble contracts, the 

 momentum acquired becomes concentrated in a smaller mass of water 

 near the bubble, and the velocity of flow in this region increases. The 

 bubble surface must then move toward the wall with increasing speed 

 as if attracted to it. This effect is so much larger than the repulsion 

 when the pressure exceeds hydrostatic that the dominant motion is an 

 apparent attraction increasing the bubble velocity toward the wall as it 

 contracts, even though the momentum of the flow is decreasing in the 

 most contracted stages. 



A free surface has the opposite effect on bubble migration, as in this 

 case the water at the surface is free to move but must do so in such a 

 way as to equalize the pressure with that of the atmosphere. As the 

 bubble begins to expand, the water above it has less inertia and is dis- 

 placed more readily than that below, and the bubble surface moves up- 

 ward. When the gas pressure falls below hydrostatic, however, a down- 

 ward flow of water takes place, because the water near the free surface 

 is more easily accelerated toward the bubble. Just as for the rigid 

 surface, the large amount of momentum acquired while the bubble is 

 large is concentrated into larger velocities of a smaller region near the 

 bubble surface as it contracts. This increased velocity makes it appear 

 that the bubble is repelled downward from the surface, and the velocity 

 increases as the size of the bubble decreases. 



8.10. Calculated Motion of a Gas Sphere near Surfaces 



A. The period of oscillation. When free or rigid infinite surfaces are 

 present, it is necessary to integrate the equations of motion derived in 

 the preceding section (Eqs. (8.52), (8.53)). These equations, or ap- 

 proximate forms of them, have been integrated numerically by several 

 writers. On the basis of such results, a number of simplifying approx- 

 imations can be made which permit an analytic expression for the period. 

 These approximations, some of which are also discussed in other sec- 

 tions, are as follows: 



(i). The period T and vertical momentum S at the first minimum 

 are twice as large as the time tm and momentum Sm at the first maxi- 

 mum. This assumption is exactly true in the limit of small vertical 

 displacements, and is very nearly true from numerical integrations. 



(ii). The vertical velocity is zero up to the first maximum. This is 

 consistent with the numerical results, which show that this velocit}^ be- 

 comes large only as the bubble radius becomes small. 



Taking dh*/dt* = 0, 6* = &«*, where h* is the initial distance of the 

 charge above the bottom, the energy equations (8.52) and (8.57) for 

 E{a*)/Y give on solving for da^/dt"^'. 



