198 10 



where c denotes the velocity of sound in water; see Equation [^8]. To make 

 this formula approximately correct, the amplitude of oscillation must be 

 large enough to make the peak of emitted pressure a sharp one, but not so 

 large that great compression of the water occurs; the range of its validity 

 may be something like 



1 /?„:„ 1 , ^ R^^^ 

 _<^< _, 1.6< --^<2.75 



A more interesting quantity is the dimenslonless ratio of the 

 energy emitted in a cycle to the total energy of vibration, E. The excess 

 energy that is present as a result of the oscillatory motion is the same as 

 the kinetic energy of the water at the instant at which R = R^, since, if 

 this energy were suddenly removed at that instant, the sphere would remain in 

 equilibrium. As the gas expands to maximum radius, this kinetic energy Is 

 expended in doing work against the difference between the hydrostatic pres- 

 sure and the pressure of the gas, and the latter work is readily calculated.* 

 In this way the energy of vibration is found to be 



provided y = U/3 ; compare Equation [49a]. Thus for the first cycle 



Q _ V2 cTq \rJ . 



ffi 3 \rJ 3 



or, after inserting c = 4810 x 12 inches per second and using Equation [2b], 



E ^ ^ 1 IR^\' _ 4 



where p^ is the hydrostatic pressure measured in atmospheres. 



Measurements of the radiated energy ^re not available, but a com- 

 parison may be made between the calculated loss by radiation and the total 

 observed loss of oscillatory energy, which is easily found from the progres- 

 sive decrease in the maximum radius for successive oscillations. From Equa- 

 tion [15]. the change in energy is 



The water is driven, so to speak, by two springs, the gas inside and the hydrostatic pressure outside. 

 As it oscillates, one spring loses energy while the other gains energy; the excess of the gain by one 

 spring over the loss by the other, as the radius changes from its equilibrium value R^ to a value R, 

 represents the potential energy of vibration. 



