150 



V. OSCILLATIONS AND CYCLIC MOTIONS. 



= + v l = 



v 



and making use of the fundamental formula of imaginaries, 

 31) e ivt = cos vt + i sin vt, 



and the principle that both the real part and the coefficient of i in 

 the imaginary part of a solution are particular solutions, we obtain 

 the two particular solutions 



e ut cosvt and 



Fig. 31. 



We thus obtain the general solution 

 32) s = e> ut (Acosvt + Bsinvf) 



(A and B being new arbitrary constants), 

 or as in 19 equation 10), 



33) 



The trigonometric factor represents a simple harmonic oscillation, 

 which on account of the continually decreasing exponential factor 

 dies away as the time increases (Fig. 32). Such a motion is called a 

 damped oscillation, and 7, is a measure of the amount of damping. 

 The extreme elongation occurs when 



34) % = a 



that is when 

 35) tan 



2 J. \ * 



X 2 'tCC] = - 



/ 1/A.lt*-.it* 



