.4 



.3- 



r9- 



cm e 

 O 



CC 



^R. [ p o//>c 2 ]' 



k = 'o 2 / 



Cc(t-to)-r 

 R 2 



Figure 10. Pressures and shocks radiated by a vapor cavity collapsing in a compressible liquid. 



(Values according to Mellen [301.) 



Mellen [30] has made the indicated calculations, taking approximate values for 

 (c + u), with the result shown in Figure 10. Each solid line exhibiting a vertical 

 rise shows the value of r<f> (very nearly equal to rp s /p) as a function of time for one 

 selected value of a parameter involving the radial coordinate. In the case of the col- 

 lapse of a cavity in water with external pressure P equal to one atmosphere, the three 

 radial distances indicated would be approximately 3.6, 360, and 36000 R x . The peak 

 pressures at the "shock" front are, for the same case, 59, 0.42, and 0.0035 atmospheres. 

 Mellen also obtained estimates of the peak pressures experimentally and found good 

 agreement with the calculations. 



From the pressures shown in Figure 10 as functions of the time, the corre- 

 sponding frequency spectra might be computed. The spectrum shown in Figure 6 

 could be corrected, at high frequencies, for the compressive effects neglected in its 

 original derivation. In view of the somewhat limited accuracy of the values shown 

 and of a number of idealizations made tacitly in the brief treatment presented above, 

 it is sufficient merely to indicate the high-frequency asymptote corresponding to the 

 discontinuity in the sound pressure. The resulting computed spectra (Figure 11) 

 do show a fair resemblance to observed spectra of cavitation noise. It appears, 

 however, that the computed magnitude of the sudden rise in pressure at the shock 

 front, relative to the parts of the wave which determine the low-frequency part of the 

 spectrum, is higher than is really the case. This is not surprising. The computations 

 assume no loss of energy in the propagated wave except that inherent in the Hugoniot 

 conditions, whereas, in fact, other losses do occur in the propagation of the high- 

 frequency components of sound waves. It is possible, also, that small amounts of 



256 



