504 Mr. N. C. Krislinaiyar on Amplitude of Vibrations 



Assuming 



y = A x sin pt H- Bj cospt + A 3 sin 2>pt 4- B 3 cos 3pt 



+ A 5 sin hpt -f B 5 cos 5pt -f- . . . . 



and equating to zero the coefficients of sin pi and cos pt in 

 the relation obtained when this value of y was substituted 

 in the differential equation, he showed that A 3 and B 3 were 

 of the order a and therefore negligible relatively to A x and 

 Bj. Hence as a first approximation, 



A 1 (n 2 -p 2 )-C^ + ^)Bi = 0, 



Therefore 



Bi 



A, 



and (n 2 —p 2 y. 



B 1 (n 2 ~p 2 )^(^-a)A 1 = 0. 



ir — p' 

 kp + a 



kp 



n 2 —p 2 



v a — kp 



V a-\- kp 



k 2 p 2 . 



According to this theory, the phase tan" 1 B ] /A 1 is con- 

 stant and independent of the amplitude maintained, and the 

 amplitude ^A^ + B^ is indeterminate. But it is known* 

 experimentally that the phase of the vibration is not inde- 

 pendent of the amplitude, and a study of the maintained 

 amplitudes brings out a number of results which the above 

 theory does not indicate. 



In the case of fork maintenance the vibration of the wire 

 reacts on the vibration of the fork and alters the amplitude 

 ■of the latter, and so a difficulty arises in interpreting the 

 results observed. The author f found that when an alter- 

 nating current of 50 cycles, and therefore of 100 heat cycles, 

 was passed through a wire whose length and tension were 

 adjusted to a frequency of 50, the wire on account of the 

 periodic thermal expansion vibrated with a large amplitude. 

 The effect of the earth's magnetic field was found to be 

 negligible, the experiment succeeding equally well in all 

 positions of the wire including the position when the wire 

 was along the direction of the earth's magnetic force. The 

 periodic thermal expansion due to the alternating current 

 heating being independent of the vibration of the wire, 

 there is no reaction between the vibrating system and the 



* C. V. Raman, Physical Review, December 1912, pp. 451-453; N. C. 

 Krishnaiyar, Physical Review, December 1919, pp. 494-496: R.N.Ghosh, 

 Proc. Indian As^oc. Cultivation of Science, vol. vi, pp. 75 & 84(1920). 



t N. C. Krishnaiyar, " On the maintenance of vibrations of wires by 

 •electric heating-," Phvsical Review, n.s. vol. vi. no. 6. 



