230 Lord Rayleigh on Maintained Vibrations, 



vibrations maintained by wind (organ-pipes, harmonium-reeds, 

 seolian harps, &c), by heat (singing flames, Rijke's tubes, 

 &c), by friction (violin-strings, finger-glasses, &c), as well as 

 the slower vibrations of clock-pendulums and of electromag- 

 netic tuning-forks. When the amplitude is small, the force 

 acting upon the body may be divided into two parts, one pro- 

 portional to the displacement 6 (or to the acceleration), the 

 second proportional to the velocity d6 / dt. The inclusion of 

 these forces does not alter the form of (1). By the first part 

 (proportional to 0) the pitch is modified, and by the second 

 the coefficient of decay*. If the altered k be still positive, 

 vibrations gradually die down; but if the effect of the included 

 forces be to render the complete value of k negative, vibra- 

 tions tend on the contrary to increase. The only case in 

 which according to (1) a steady vibration is possible, is when 

 the complete value of k is zero. If this condition be satisfied, 

 a vibration of any amplitude is permanently maintained. 



When k is negative, so that small vibrations tend to increase, 

 a point is of course soon reached after which the approximate 

 equations cease to be applicable. We may form an idea of 

 the state of things which then arises by adding to equation (1) 

 a term proportional to a higher power of the velocity. Let 

 us take 



s+«s «©■«■«-». ■ ■■•.■ <*> 



in which k and k! are supposed to be small. The approximate 



solution of (2) is 



K f nk z 

 6 = Asuint-\ ^—cosSnt, .... (3) 



in which A is given by 



* + f«V 2 A 2 =0 (4) 



From (4) we see that no steady vibration is possible unless k 

 and k! have different signs. If k and k! be both positive, the 

 vibration in all cases dies down ; while if k and k' be both 

 negative, the vibration (according to (2)) increases without 

 limit. If k be negative and k! positive, the vibration becomes 

 steady and assumes the amplitude determined by (4). A 

 smaller vibration increases up to this point, and a larger vibra- 

 tion falls down to it. If, on the other hand, k be positive, while 

 K f is negative, the steady vibration abstractedly possible is 



* For more detailed application of this principle to certain cases of 

 maintained vibrations, see Proceedings of the Koyal Institution, March 15, 

 1878. 



