156 V. OSCILLATIONS AND CYCLIC MOTIONS. 



when the second member is equal to zero, or in physical terms the 

 forced and free vibrations exist superimposed. Accordingly we have 



If the system starts from rest we must determine A and /3 so that 

 when t = 0, s and -^- are equal to zero. These conditions will be 

 very nearly satisfied, if p and h are nearly equal and x small, by 



( i ^ 



I ~~9 y ' t (-\ / x 2 Yl 



52) s = a\cos (pt a) e cos ( J/ft 2 t a \ r 



The simultaneous existence of two harmonic vibrations of nearly 

 equal frequencies gives rise to the phenomenon known as beats. 

 Suppose 



53) s = a cos (pt a) + & cos {(p + Ap] t /3), 



where Ap is a small quantity, equal to 2 x times the difference of 

 frequencies. We may write the last term 



& cos {(pt a) + dp - t -f a /3) 



= 6 {cos (pt a) cos (dp t + a /3) 



- sin (pt a) sin (dp -t-\- a /3)}, 

 so that 



54) s = {a -f- & cos (Ap t + a /5)} cos (pt a) 



- & sin (dp - 1 -f a /3) sin (^^ a), 

 or if we write 



a -f & cos (z/# -f- a /3) = A 



- & sin (z/p t -f a /3) = 5, 



55) 5 = Dcos(pt a f), 

 where 



and 



D = 



Accordingly the compound vibration may be considered as a harmonic 

 motion of variable amplitude and phase, the amplitude varying from 



a + ~b to a &, with the period -^- and frequency -^- equal to the 



difference of the frequencies of the two constituents. The phenomenon 

 of beats or interferences is represented graphically in Fig. 34. 



