Cambridge have revealed great difficulties in trying to detect whether or not shock 

 waves are present in cavitation pressure fluctuations, but left no doubt that they do 

 arise at least sometimes. It does not seem untimely to express a hope that in the hands 

 of able experimenters such as Professor Knapp and Dr. Ellis, this aspect of cavitation 

 damage will soon be fully elucidated. 



The second reason justifying attention to the matter of shock waves concerns 

 the cavitation noise spectrum, as is found by making frequency analyses of hydrophone 

 outputs. It has been pointed out before now that if a finite pulse possesses a dis- 

 continuity of the sort represented by the mathematical property f(t + ) — f(t~) ^L (i.e., 

 a finite "jump" as at a shock front), its energy spectrum is asymptotically proportional 

 to the inverse square of frequency; in other words, it decreases at a rate of 6 decibels 

 per octave. Moreover, no other kind of pulse has a spectrum with this asymptotic 

 property. This fact may possibly have a bearing on some experimental measurements 

 of noise spectra; but a probably more useful consideration is as follows. If a pulse 

 having the property described above is applied to a resonant system, the frequency 

 response decreases asymptotically (i.e., well above the resonance frequency) as the 

 inverse fourth power of frequency. Again, no other kind of pulse produces this response. 

 Thus the response at very high frequencies from a practical hydrophone, which is 

 bound to be affected by self-resonances at a number of frequencies, should be sus- 

 tained at a slope of — 12 decibels per octave if shocks are present. 



The theoretical problem of shock formation by collapsing bubbles is clearly a 

 very difficult one. The allied but evidently simpler problem where a spherical shock 

 arises as the result of conditions prescribed over a spherical boundary, such as Sir 

 Geoffrey Taylor, Dr. Whitham and many others have treated, seems formidable enough; 

 but the cavitation problem presents the great additional difficulty that the motion of 

 the inner boundary (the bubble surface) is not initially known. Dr. Gilmore, for 

 instance, has made an important contribution towards the understanding of bubble 

 motion in compressible fluids, and his method of treatment was the first to give useful 

 results for cases where the fluid velocities are of the order of the sound velocity; but 

 apparently much remains to be done to clear up the question of strong cavitation where 

 a shock wave probably forms very close to the bubble surface and has a profound 

 effect on the bubble motion. 



I understand that Mr. Fitzpatrick has made some calculations, not mentioned 

 explicitly during his talk, which enabled him to estimate the least severe conditions of 

 cavitation which should give rise to shock waves, and independently I have also 

 attempted this. My own work stopped short after considering a weak shock formed 

 far from the cavity centre, but I hope eventually to hear that Mr. Fitzpatrick has pro- 

 ceeded closer to the "heart of the matter." 



These remarks are intended only to reaffirm the desirability of regarding water 

 as a compressible fluid as far as cavitation collapse is concerned, and so to commend 

 compressibility effects in cavitation as an interesting and useful field of study. Recog- 

 nition of the theoretical difficulties need not reflect a pessimistic view of progress in 

 this subject, for the signs are that a great deal will be achieved towards a complete 

 solution within the next few years. 



F. R. Gilmore 



The authors have given a very interesting and well organized review of the 

 problem of hydrodynamic noise. I wish to comment only on the section dealing with 

 the noise produced by a collapsing cavity. According to the theoretical work of Lord 

 Rayleigh, as a spherical bubble in an incompressible liquid collapses to infinitesimal 

 size, velocities and pressures in the neighborhood of the bubble approach infinity, 

 provided that the pressure of any vapor or gas in the bubble either remains constant 

 or at least does not rise rapidly enough to prevent complete collapse. Recent 

 theoretical work yields similar results even when the compressibility of the liquid is 



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