THEORY OF VIBRATIONS 25 



and obtain, on substitution, 



We have now three cases to distinguish. If the friction be 

 relatively small, more precisely if k < 2n, we may put 



r^ = 7i 2 -J#, ..................... (6) 



and the solution of (3) is 



y = A cos n't + B sin n't, ............... (7) 



whence x = e " **' (A cos n't + B sin n't) ............. (8) 



Changing the arbitrary constants, and putting 



r = 2/&, ........................... (9) 



we have x = ae~^ T cos(n' + e) ................ (10) 



This may be described as a modified simple-harmonic vibration 



in which the amplitude (ae ~ ^ r ) sinks asymptotically to as t 

 increases. The time T in which the amplitude is diminished 

 in the ratio l/e is called the " modulus of decay." The 

 relation between x and t is exhibited graphically in Fig. 11, 



Fig. 11. 



where the dotted lines represent portions of the exponential 



curves x = ae ~ . For the sake of clearness the rapidity 

 of decay is here taken to be much greater than it would be in 

 any ordinary acoustical example. 



