158 THE ROYAL SOCIETY OF CANADA 



for a tension in a fluid medium to exist, and that therefore cavitation 

 will set an upper limit to the amount of energy which it is 

 possible to transmit. On the other hand, others think that in the 

 audible and higher ranges of frequency the vibrations are so rapid 

 that there is not time for discontinuities to be formed in the medium, 

 even if the alternating pressure in the wave exceeds the static pressure ; 

 that the medium can, during the time of half a vibration, be under a 

 tension; and that with increasing rapidity of vibration the medium 

 will behave more and more like an elastic jelly. Cavitation of the 

 kind described above would therefore never be encountered. 



In another connection Stokes made a remark which would seem 

 to apply here: "... When a body is slowly moved to and fro in any 

 gas, the gas behaves almost exactly like an incompressible fluid, and 

 there is merely a local reciprocating motion of the gas from the 

 anterior to the posterior region, and back again in the opposite phase 

 of the body's motion, in which the region that had been anterior 

 becomes posterior. If the rate of alternation of the body's motion 

 be taken greater and greater, or, in other words, the periodic time less 

 and less, the condensation and rarefaction of the gas, which, in the 

 first instance, was utterly insensible, presently becomes sensible, and 

 sound waves (or waves of the same nature in the case the periodic 

 time be beyond the limits of audibility) are produced, and exist along 

 with the local reciprocating flow. As the periodic time is diminished 

 more and more of the encroachment of the vibrating body on the 

 gas goes to produce a true sound wave, less and less a mere local 

 reciprocating flow. . . ."^ 



If cavitation arising from a tension in the medium were to be 

 encountered in a fluid, it is possible to calculate the maximum ampli- 

 tude of displacement, at any frequency, for the transmission of 

 energy and the maximum energy transmissible. 



Let p = the density of the medium; c, the velocity of sound in it; 

 /, the frequency; a, the amplitude of displacement; and p the ampli- 

 tude of alternating pressure in the waves. 



Then, considering only the case of plane waves, p = 2Tfpac, all 

 quantities being in C.G.S. units. 



If the static pressure of the medium at the point considered 

 equals po, then if cavitation can take place p cannot exceed po- 



Therefore the maximum amplitude for transmission, at the stated 



Po 



frequency/, is ^ , from which it can be seen that for any constant 



ZTjpC 



iStokes, Mathematical and Physical Papers, Vol. IV, p. 299. 



