20 DYNAMICAL THE OB Y OF SOUND 



For example, let 



* ..................... < 16 > 



this represents a force which is sensible for a greater or less 

 interval on both sides of the instant t = 0, according to the 

 value of r, the integral amount or impulse being //,*. By 

 making r sufficiently small we can approximate as closely as 

 we please to the case of an instantaneous impulse. Since 



cosntdt_7r r smntdt_ m 



~ ~ 



LL6 ** T 



we have x= - sin nt ................... (18) 



The exponential factor shews the effect of spreading out 

 the impulse. This effect is greater, the greater the frequency 

 of the natural vibration. 



9. Forced Oscillations in any System with One Degree 

 of Freedom. Selective Resonance. 



The generalization of these results offers no difficulty. When 

 given extraneous forces act on a system with one degree of 

 freedom, whose coordinate is q, the work which they perform in 

 an infinitely small change of configuration, being proportional to 

 8q, may be denoted by QSq. The quantity Q is called the 

 "force" acting on the system, "referred to the coordinate q." 

 For instance, if q be the angular coordinate of a body which can 

 rotate about a fixed axis, Q is the moment of the extraneous 

 forces about this axis. 



It follows that in any actual motion of the system the rate 

 at which extraneous forces are doing work is Qq. The equation 

 of energy now takes the form 



j t (T+V)=Qq, ..................... (1) 



whence, inserting the value of T from 7 (1), we have 



' 



* The graph of this function is given, for another purpose, in Fig. 14, p. 33. 

 t The former of these integrals is evaluated in most books on the Integral 

 Calculus. 



