Internal Friction on Resonance. 179 



Let 



P = sin /ml . 



f**r § + € -m*Z.i 



(4 = COS fit . . 



>.. . (10) 



J 



{; 



If possible, let m be other than n; when #=/, we have <£ = 

 and -v/r = 0, or 



A 1 P + B 1 Q=0, 

 .B i P-A 1 Q=0; 



therefore, since A v B l must be real, they must vanish, and we 

 conclude that the only steady vibration is of the same period as 

 that impressed on the extremity. 



Let m = n] when x — l, <f> = A and-^ = 0; hence 



PAj + QB^A,! 

 PB,-QA ,=();/ 



AP 



(ID 



A l~ p'2 + Q 



T\ - AQ 



^~P 2 + Q 2 



This completely determines the steady vibration of the string. 



Suppose a change to take place in the forcing vibration, it is 

 easy to see that the result will be that momentarily all the notes 

 natural to the string with both ends fixed will be sounded. This 

 conclusion could readily be tested by graphically describing the 

 motion of a point of a string moving in the manner supposed, 

 the motion being produced by a tuning-fork actuated by an 

 electromagnet. If this be verified, an attempt might be made 

 to determine the value of k for various strings or wires by com- 

 paring the amplitude of vibration at the points of greatest and 

 least vibration ; and at the different points of least vibration 

 true nodes will not occur. The curve having x for abscissa, and 

 the maximum value of f at each point for ordinate, might pos- 

 sibly be portrayed by photographing a vibrating string. The 



calculations would be much facilitated by the fact that /x=- if 



small quantities of the second order are neglected. Suppose 

 that yd—^iTy a case of strong resonance; then P = and Q=7rA7i 



very nearly; we have Aj=0 andBj= — — , and the motion is 



N2 



