44, 45] 



PHENOMENON OF BEATS. 



157 



In the case of free and forced vibrations coexisting [equation 52)], 

 we have at the beginning beats which gradually die away owing to 



the factor e 2 in the free vibration, leaving only the forced vibration. 



Fig. 35. 



Fig. 34. 



This is shown in an interesting manner by a tuning fork electrically 

 excited by another fork not quite in unison with it, the phenomenon 

 of a single driven 

 fork apparently 

 producing beats 

 with itself being 

 very striking (Fig. 

 35). It will be 



noticed that the first maximum is greater than the steady amplitude. 

 The greater part of this section and the preceding is taken 

 from Rayleigh's Theory of Sound. 



45. General Theory of small Oscillations. Having now 

 set forth the general characteristics of vibrations excuted by systems 

 possessing one degree of freedom, we will now treat the problem of 

 the small vibrations of any system about a configuration of equili- 

 brium after the manner of Lagrange, who first investigated it. 



Suppose a system is defined by n parameters q i9 q 2 , . . . q n - Its 

 potential energy will depend only on the coordinates q, and developing 

 by Taylor's Theorem, 



56) W = TF 



where the suffix zero denotes the value when all the g's are zero. 

 Suppose that this is a configuration of equilibrium, then W is a 



. . . (dW\ -, m, TT;r TT; r 



minimum or maximum and every (-~ I equals zero. Inus W - rK 



begins with a quadratic function of the #'s. If the motion is small 

 enough we may neglect the terms of higher orders of small quantities. 

 Accordingly, neglecting the constant W (for the potential energy 

 always contains an arbitrary constant which does not afPect the 

 motion), we shall put W a homogeneous quadratic function of the #'s 

 with constant coefficients, 



r n s = n 



57) W= 



