16 DYNAMICAL THEOBY OF SOUND 



point of suspension in the unstrained state, the kinetic energy 

 is given by 



..................... (12) 



if p be the line-density, I the unstretched length, and q the 

 displacement of the weight. The inertia of the spring can 

 therefore be allowed for by imagining the suspended mass to be 

 increased by one-third that of the spring. 



8. Forced Oscillations of a Pendulum. 



The vibrations so far considered are " free," i.e. the system 

 is supposed subject to no forces except those incidental to its 

 constitution and its relation to the environment. We have 

 now to examine the effect of disturbing forces, and in particular 

 that of a force which is a simple-harmonic function of the time. 

 This kind of case arises when one vibrating body acts on 

 another under such conditions that the reaction on the first 

 body may be neglected. 



For definiteness we take the case of a mass movable in a 

 straight line, the subsequent generalization ( 9) being a very 

 simple matter. The equation (1) of 6 is now replaced by 



(1) 



the last term representing the disturbing force, whose amplitude 

 F, and frequency p/Zir, are regarded as given*. If we write 



f, .................. (2) 



we have -j+n?x=fcospt ................... (3) 



The complete solution of this equation is 



x = A cos nt + B sin nt 4- -~ - cos pt t ...... (4) 



7i 2 p z 



as is easily verified by differentiation. 



The first part of this, with its arbitrary constants A } B, 

 represents a free vibration of the character explained in 5, 



* The slightly more general case where the force is represented by F cos (pt + a) 

 can be allowed for by changing the origin from which t is reckoned. 



