Lord Rayleigh on Maintained Vibrations. 231 



unstable, a departure in either direction from the amplitude 

 given by (4) tending always to increase. 



Of the second class the vibrations commonly known as 

 forced have the first claim upon our attention. The theory of 

 these vibrations has long been well understood, and depends 

 upon the solution of the differential equation formed by 

 writing as the right-hand member of (1) ¥ cos pt in place of 

 zero. The period of steady vibration is coincident with that of 

 the force, and independent of the natural period of vibration ; 

 but the amplitude of vibration is greatly increased by a near 

 agreement between the two periods. In all cases the ampli- 

 tude is definite and is proportional to the magnitude of the 

 impressed force. When the force, though strictly periodic, is 

 not of the simple harmonic type, vibrations may be maintained 

 by its operation whose period is a submultiple of the principal 

 period. 



There is also another kind of maintained vibration which 

 from one point of view may be considered to be forced, inas- 

 much as the period is imposed from without, but which differs 

 from the kind just referred to in that the imposed periodic 

 variations do not tend directly to displace the body from its 

 configuration of equilibrium. Probably the best-known ex- 

 ample of this kind of action is that form of Melde's experiment 

 in which a fine string is maintained in trans \erse vibration by 

 connecting one of its extremities with the vibrating prong of 

 a massive tuning-fork, the direction of motion of the point of 

 attachment being parallel to the length of the string*. The 

 effect of the motion is to render the tension of the string 

 periodically variable ; and at first sight there is nothing to 

 cause the string to depart from its equilibrium condition of 

 straightness. It is known, however, that under these circum- 

 stances the equilibrium position may become unstable, and 

 that the string may settle down into a state of permanent and 

 vigorous vibration, whose period is the double of that of the 

 point of attachment^. 



The theory of vibrations of this kind presents some points 

 of difficulty, and does not appear to have been treated hitherto. 

 In the present investigation we shall start from the assumption 

 that a steady vibration is in progress, and inquire under what 

 circumstances the assumed state of things is possible. 



If the force of restitution, or " spring," of a body suscep- 

 tible of vibration be subject to an imposed periodic variation, 



* When the direction of motion is transverse, the case falls under the 

 head of ordinary forced vibrations. 



t See Tyndaffs ' Sound/ 3rd ed. ch. iii. § 7, where will also be found 

 a general explanation of the mode of action. 



S2 



