193 



The maximum radius /J^, and the mihlmum radius R^^^ are connected 

 by the equation 



R„,„ ^ 3\rJ - 3\rJ ^ /?_ ^5a. 



•^n.... 



which for large amplitudes can be shortened approximately to 



R. i/ie_N^^_^ j^^j 



^ max 



R^:„ ~ 3\ R^ I ' R 



see Equation [30], with y = k/j>. 



It Is noteworthy that, as the amplitude Increases, the time- 

 displacement curve, which approximates to a sine curve at small amplitudes, 

 becomes more and more pointed near the minimum radius. Thus the sphere 

 spends very little time at radii below the equilibrium radius Rq when the am- 

 plitude is large. This effect arises physically from the diminution in the 

 area across which the inrushlng water is moving; because of this diminution 

 the water has a strong tendency to increase in velocity, and hence the gas 

 meets great difficulty in stopping the motion. 



The curves are calculated on the assumption of incompressible water. 

 For this reason the incoming and outgoing motions as shown in Figure 2 are 

 similar. When the minimum radius becomes extremely small relatively to R^, 

 however, compression of the water begins to play a role, as already stated; 

 consequently, the amplitudes of successive oscillations will progressively 

 decrease. Each loop of the actual curve, extending from one minimum to th - 

 next will be very nearly the same as it would be, at the same maximum radius, 

 for incompressible water. 



PRESSURE IN THE WATER 



Let the pressure in the water at great distances be the hydrostatic 

 pressure p^; and let gravity be assumed not to act. At the surface of the 

 sphere of gas, the pressure in the water must be that of the gas or 



/R \^y 

 ^ = (?) Po [6] 



by Equation [26], where R is the instantaneous radius of the sphere. At any 

 other point, at a distance r from the center of the sphere, the pressure, if 

 the motion is non-compressive, is found to be 



P = f (p. + 2^'""' ~ Po) - 2 ^^' + Po t7a] 



