MOTION OF THE GAS SPHERE 331 



required in bringing the charge (source) and its image to their actual 

 separation from infinite distance. This work is then the increase in 

 kinetic energy, which is positive for hke charges (rigid surface) repelUng 

 each other and negative for unUke, attracting charges (free surface). 

 ExpHcit calculations using the image force =be^/4a:^, where x is the 

 separation, leads directly to the results given. 



The hydrodynamical reason for the changes in kinetic energy is 

 simply that for a free surface there is iess inertia of surrounding fluid 

 than in an infinite medium, while a rigid boundary interferes with the 

 flow and increases its inertia. As Herring (46) points out, this expla- 

 nation also makes obvious the observed changes in period. The effect of 

 either surface is to change the inertia (or kinetic energy) of the oscil- 

 lating system, without affecting its spring constant (or potential 

 energy). The period is proportional to the square root of the inertia, 

 as in the formula for a mass oscillating on a spring, and the percentage 

 change in period is therefore approximately one-half the change in 

 kinetic energy. Hence we might expect the modified periods Ts, Ti to 

 be given by 





where T is the free water period, and aav is the time average of a over 

 one complete cycle. (The justification for using a^v as defined is not 

 obvious, but a detailed argument, such as Herring has given, is omitted, 

 as more exact derivations are given in section 8.10.) The decreased 

 inertia for a free surface thus shortens the period and a rigid surface 

 causes an increase. 



The presence of rigid and free surfaces also causes displacements of 

 the bubble center as it pulsates, and these changes can be qualitatively 

 understood in terms of the change in inertia of the flow. In the case of 

 a rigid surface, the presence of the boundary interferes with radial flow 

 of water, whether outward or inward, near a spherical surface in its 

 vicinity. Initially, when the pressure in the gas is in excess of the 

 hydrostatic pressure, the water on the side of the bubble surface near 

 the wall is less readily displaced, and the bubble surface moves away 

 from the wall. The effect is relatively small, however, because the net 

 pressure (in excess of hydrostatic) is positive for a short part of the 

 bubble period, and the bubble is small during this time. When the 

 pressure falls below hydrostatic, acceleration of flow toward the bubble 

 surface does not occur as readily on the side toward the wall, and the 



