Rayleigh [18], in which (p p — P) is constant and equal to P . He found, by express- 

 ing the constancy of the sum of the kinetic energy, l^pR^R 2 , and the potential energy, 

 4 

 -ttR 3 P , that the radius and the wall velocity are related by the equation, 



(R h /R) s = 1 + (3 P £ 2 /2Po). (13) 



For this case, R x is the maximum radius of the cavity. 



Some cavities rebound after collapse. The mechanism by which the rebound 

 comes about is not known. Apparently a small quantity of gas contained in the cavity 

 plays an essential role, but details are uncertain. What is known is that the flow 

 velocity and pressures attain such values that the acoustic theory is not applicable to 

 the part of the motion for which the cavity is very small. Figure 5 continues the 

 illustration, however, showing the behavior typical of those cases in which rebound 

 occurs. The assumption of an empty cavity and an incompressible liquid have been 

 retained. The cavity is assumed to collapse to an indefinitely small radius and to 

 rebound with an arbitrarily postulated loss of energy. The whole sequence is clearly 

 only an illustration; actual cavities show a wide range of behavior. Knapp and Hol- 

 lander [13] photographed cavities which rebounded as many as five times. Benjamin 

 [19] has also observed repeated rebounds under different conditions. 



r 2 S 



/>r;p 



f R, \P/o 



Figure 6. Spectral distribution of the sound, for motions shown in Figure 5, as computed for an 



incompressible liquid. 



The spectrum of the sound: acoustic theory. — From the sound pressure, the 

 spectral distribution of the radiated energy may be computed. Figure 6 shows the 

 spectra corresponding to the growth and single collapse and to the growth and multiple 

 collapse postulated in the example. The spectra exhibit maxima at frequencies of the 

 order of the reciprocal of the time required for growth and collapse. At lower fre- 



251 



