Waves generated by Impact. 97 



early part of the collision is given by an expression of the 

 type 



4>= — ~ T cos nt + T , 



4 a 2 nk \ 4 / 



where a is the relative velocity of impact, a 2 is a certain 

 constant which can be easily calculated from Lamb's theory, 

 and 



7 4 y^.E „ /E -857T 



r being the radius of the sphere, E the Young's modulus, 

 and p the density. 



The leading term due to the end of the collision is obtained 

 from this by changing nt to n(t — r), r being the duration of 

 impact. 



Also the ratio of the maximum kinetic energy of vibrations 

 to the energy before collision is approximately given by an 

 expression of the type 



R= 50' VWp) 



Since v^E/p is the velocity of longitudinal vibrations along 

 a bar of the material of the solids in question, we see that, 

 in general, the expression for <f> is very small in magnitude, 

 and that R is an exceedingly small ratio. 



Lord Rayleigh's results show that under ordinary con- 

 ditions, that is, unless the spheres are very large in size or 

 the relative velocity of impact is very great, vibrations 

 should not be generated in appreciable degree, and that the 

 energy of the colliding spheres remains translational. More- 

 over, even if vibrations be excited at all, the pitch of the 

 gravest sound so produced would be very high, in fact almost 

 beyond the range of audibility. For example, in the case of 

 two mahogany balls of 6 cm. diameter, the frequency of the 

 gravest vibrations excited would be about 37,000 per sec. 

 We know, however, from experience that when two spheres, 

 say two billiard-balls, impinge directly upon each other, aerial 

 waves of considerable intensity are generated which are 

 audible as the characteristic sound of impact. The investi- 

 gation described in the present paper was undertaken to 

 ascertain, both theoretically and experimentally, the origin 

 and characteristics of the sound produced by such impact. 



Since, as we have seen, under ordinary conditions vibrations 

 cannot be excited in any perceptible degree, practically the 



Phil. Mag. S. 6. Vol. 32. No. 187. July 1916. H 



I 



