80 DYNAMICAL THEORY OF SOUND 



28. Forced Vibrations of a String. 

 The simplest case of forced vibration is where a given 

 simple-harmonic motion 



y = 0coa(pt + a) .................. (1) 



is imposed at a point (x = a). The portions of the string on the 

 two sides of this point are to be treated separately. The 

 results are 



. px 

 sin 



cos (pt -fa) [0 < x < a] , 



sm 



sn 



for these satisfy the general differential equation 22 (2), they 

 make y 1 = for x = 0, and y z = for x = I, and they agree with 

 (1) when x = a. The amplitude of y l or y 2 becomes very great, 

 owing to the smallness of the denominator, whenever pa/c or 

 p(l a)/c is nearly equal to a multiple of TT, i.e. when the 

 imposed period 2ir/p approximates to a natural period of a 

 string of length a or I a, respectively. To obtain a practical 

 result in such cases we should have to take account of dissipative 

 forces. 



The case is illustrated by pressing the stem of a vibrating 

 tuning fork on a piano string. The sound swells out powerfully 

 whenever the portion of the string between the point of contact 

 and either end has a natural mode in unison with the fork. 

 This plan is recommended by Helmholtz as a means of producing 

 pure tones, since the higher modes of the fork, not being 

 harmonic with the fundamental, are not reinforced. 



When a transverse force of amount Y per unit length acts 

 on the string, the equation (2) of 22 is replaced by 



In general Y will be a function both of x and t. 



The case of a periodic force F cos (pt + a) concentrated on an 

 infinitely short length of the wire at x a may be deduced from 



