516 . Mr. C. V. Raman on the Maintenance of 



We may, to fix our ideas, consider a case in which the 

 frequency of the oscillations of the system is large compared 

 with the frequency of the imposed variation of spring. It is 

 clear in this case that the successive oscillations of the 

 system are not executed under identical conditions. At one 

 epoch the restitutional coefficient is a maximum, and an 

 oscillation would evidently be executed by the system during 

 this epoch in a shorter time than at the epoch when the 

 spring is a minimum, and the amplitude of the oscillation 

 would also be greater at the latter epoch. The motion of the 

 system in such a case would evidently be similar to that in 

 the atmospheric beats produced by two simple tones one of 

 which has a greater frequency and amplitude than the other 

 (Appendix XIV. of Helmholtz's ' Sensations of Tone '). 

 We have therefore to modify the expression for the displace- 

 ment applicable in the general case and may assume it to be 

 approximately represented by two terms, one of which 

 has a much larger amplitude than the other and has a 

 frequency in excess of it by that of the imposed variation of 

 spring. 



We may therefore put </> the displacement equal to 



C r cos (rnt\2 4- e r ) + C r _ 2 cos (r — 2nt/2 + e r _ 2 ). 



The expression for the restoring force at any instant requires 

 corresponding modification. The product of the variation 

 of spring and the displacement is 



— 2aw 1 2 sin w£ [ C r cos (rnt/2 -f e r ) + C r _ 2 cos (r — 2nt/2 -fe r _ 2 )]. 



It is readily seen that this has a periodic component of 

 frequency rN/2 which is equal to 



— an 1 2 G r _ 2 sin (rntj2 + e r _ 2 ) if r > 2, 

 or — 2an 1 2 G r _ 2 sinnt cos e r _ 2 if r = 2. 



The work done by this the " impressed " part of the restoring 

 force during a period of the variation of spring is readily 

 shown to be 



— u/Arn 1 2 nC r C r _ 2 tcos(€ r — e r _ 2 ), if r>2 

 or — un ] 2 n(j r G r _ 2 t cos e r cose,. _ 2 , if r = 2. 



The energy dissipated during the same time is 



iK?i 2 n 1 [r 2 C r 2 +(r-2) 2 C r _ 2 2 ]t. 

 Since the 2nd term within the square brackets is the square 

 of a small quantity it may be neglected, and the energy 

 supplied is equal to the energy dissipated if 



/miC r = — a/i l O,._ 2 cos (e r — e r _ 2 ) 

 provided r> 2, or if 



KrnO r = — 2 un 1 Cr_ 2 cos 6 r cos €r_ 2 when r = 2. 



