Maintenance of Vibration by Periodic Field of Fore. 133 



positive. The complete expression for the force, which may 

 be assumed to act at the point j? of the wire, is thus 



n — jo i 2 77- \ 



(1 + by + c# 2 -f &c.) 2 a n cos / ~ rj , e n y 



We may now consider, first, the ordinary forced vibration. 

 This may be obtained by the method of successive approxi- 

 mations. To begin with, we may neglect the quantities 

 by Q , cy 2 , &c, in the expression for the maintaining force, 

 which then assumes the simple form ^a n cos (2irrnt/T — e n ). 

 Since the forced vibration is of negligible amplitude, except 

 when the period of the field is more or less nearly equal to 

 one of the free periods of the wire, the harmonic components 

 in the motion may be determined, term by term, from the 

 corresponding components of the impressed force. The 

 forced vibration may therefore be written as 



n %*° 7 • nirx . niTXr, [2nirrt A 



X a n kn sm bid -cos I -7^ -e n — e n I? 



ra =l a a \ } 1 



where a is the length of the string or of each vibrating 

 segment, and hi, e n ' are quantities which, in respect of each 

 harmonic, may be expressed in terms of the natural and 

 impressed frequencies of vibration and of the decrement of 

 the free vibrations. If a is equal to a or any multiple 

 of it, the forced vibration becomes negligibly small, the 

 periodic force having an inappreciable effect when applied 

 at a node. 



An interesting example in which the formula given above 

 may be applied is that of a single impulse acting at the 

 point x Q j once in each period of vibration. The coefficient a n 

 is then the same for all the harmonics, and e n — for all 

 values of n. 



It may readily be shown that if the period of forced 

 vibration in this case is somewhat greater or somewhat less 

 than the period of free vibration of the string, the form of 

 the maintained vibration is practically the same as that of 

 a string plucked at the point x Q . For the phase-con- 

 stants ej are then practically ail equal to zero and it re- 

 spectively. Further, k n is then practically independent of 

 the dissipation of energy (whatever this may be due to), and 

 is inversely proportional to the difference between the 

 squares of the natural and impressed frequencies. For 

 different harmonics, kn is proportional to 1/n 2 , and the 



