﻿222 Mutual Influence of Resonators. 



The equation oh' motion for p ± is accordingly 



and that for p 2 differs only by the interchange of p 1 and p. 2 . 

 Assuming that both p 1 and p 2 are as functions of the time 

 proportional to e int , we get to determine n 



?l 2(M' - 47rar 3 . ikr} -fi= ± n 2 . -^Trar'R 



ikn 



or approximately 



» = \/M-{ 1+ S( ;ffi± ^ E )}- • (M) 

 If, as before, we take n—p + iq, 



p= Vw •( 1± ""tIM^ cosffi )' * * ^ 55) 



S=P-^-(^+sin*R) (56) 



We may observe that the reaction of the neighbour does not 

 disturb the frequency if cos£R = 0, or the damping if 

 sin/jR=0. When JcR is small, the damping in one altern- 

 ative disappears. The two vibrators then execute their 

 movements in opposite phases and nothing is propagated to 

 a distance. 



The importance of the disturbance of frequency in (55) 

 cannot be estimated without regard to the damping. The 

 question is whether the two vibrations get out of step while 

 they still remain considerahle. Let us suppose that there is 

 a relative gain or lo^s of half a period while the vibration 

 dies down in the ratio of e : 1, viz. in the time denoted 

 previously by T, so that 



il\— P-2)T = TT- 



(ailing the undisturbed values of p and q respectively P and 

 Q, and supposing IR to be small, we have 



P 477<7>> 4 



Q KM'~ ~ 7r ' 



in which Q l /I*=2'ir<rkr A /M!. According to this standard the 

 disturbance of frequency becomes important only when 

 kR < 1/ir, or R less than Xjir 2 . It has been assumed through- 

 out that r is much less than R. 

 Terling Place, Witham. 



