Sound- Waves of Finite Amplitude. 329 



which gives us a finite upper limit to the time required for 

 discontinuity to set in, provided B is finite. As our assump- 

 tions only remain approximately true so long as the motion is 

 continuous, (15) will only give an approximation to the time 

 when discontinuity first commences, and accordingly the 

 relation must be taken to refer to that part of the wave for 

 which its right-hand side is a minimum. If B is negative 

 (which is not the case for any known substance), the appro- 

 priate part of the disturbance will be such that 'du/'ftr is 

 positive. 



To determine approximately the value of B, we may refer 

 to (13) and the inequality immediately following. If we 

 assume Boyle's law of pressure, so that V {dp/dp) = const., we 

 have evidently 



B = 1 very nearly. 



If we assume that the changes of density take place adia- 

 batically, so that/) oc/> v and y is nearly constant, the approxi- 

 mate value of B becomes 



1+ I\Z|-"V # 



by means of (11) ; 



dp V dp V dp 



_7 + l 



If, then, viscosity be neglected, we must conclude that under 

 any practically possible law of pressure the motion in spherical 

 sound-waves always becomes discontinuous, and a fortiori the 

 same will be true of cylindrical waves. But inasmuch as 

 our result for spherical waves depends on the existence of an 

 infinite logarithm in (14) when t± is increased without limit, 

 we may conclude that for waves diverging in tour dimensions 

 (or, more generally, in any number of dimensions finitely 

 greater than three) there would be some cases where the 

 motion remained always continuous. 



11. The general question of spherical sound-waves of finite 

 amplitude is by no means an easy one. In the case of plane 

 waves we can write down at once from Riemann's equations 

 the condition that the disturbance may be propagated wholly 

 in the positive or wholly in the negative direction. The 

 respective conditions are * : — 



u=± $yp h8p ' 



where p is the density of that part of the fluid whose velocity 

 * Cf. also Lord Rayleigh, 'Theory of Sound/ vol. ii. p. 35 (3). 



