Ripple Formation in Kundt's Tube. 481 



Extension of Konig's Tlieovy. 



This method of deducing the forces from Kirchhoff's 

 general equations admits of an extension which may be of 

 some use. Konig's investigation was limited to the case 

 when the two spheres experienced the same velocity. A 

 case has arisen * when it would be of value to remove this 

 restriction, and this has been done in the following. 



In KirchhofPs general equations we will assume as before 

 that Ui = v 1 = u 2 = v 2 = 0, but that w^ is different from ic 2 . Then 

 the equations for the Z forces are 



<l 11=^+2,; i3! a W z , 



dt ~dn l ~d<'i dt ~&w 2 ~dc 2 



and the energy equation is 



o 6 " oc 



+ imw£ + J »nn 2 2 . 



Now as T is e> . xplicitly in terms of iv u w 2y c u c 2 , 



we have 



Further, if W\ and w 2 are simple periodic functions of the 

 time (cos27r-J, and we wish to find the mean value of Z, 

 the only terms which need be retained are those involving 



sin 2 27r- orcos 2 27r-. 



T 



Hence 



Z 1 = —Mean value of 



* 3i I 



If we now suppose that the velocities iv 1 and iv 2 vary along 

 the tube as well as with the time according to the law, 



t ire 



iv=iv cos lit- cos ; 



'IV 



* Robinson, loc. dt. 

 Phil Mag. S. 6. Vol. 19. No. 112. April 1910. 2 I 



