30 



DYNAMICAL THEORY OF SOUND 



(within the limits of a quarter-period) the instant of maximum 

 velocity. Thus if when the particle is at P, on its way to 0, 

 the velocity be increased in the ratio of PQ to PQi, the phase 

 is accelerated by the angle QOQ l} whilst a similar impulse at P' 

 would retard the phase by the angle Q'OQ\. 



In order that no effect may be produced on the phase it 

 is necessary that the impulse be delivered at the instant of 

 passing through 0. If we imagine that a small assisting 

 impulse is given at every such passage, as in the case of the 

 ordinary clock escapement, we have an illustration of the 

 circumstances of maxi- 

 The 



mum resonance, 

 period of the disturbing 

 force is exactly equal to 

 the natural period, and 

 the force synchronizes 

 with the velocity. The 

 amplitude is deter- 

 mined by the considera- 

 tion that the work done 

 by the impulses must 

 balance that lost by 

 friction. The result is 

 not essentially different 

 if the impulse be dif- 



Fig. 13. 



fused symmetrically about 0, as in the case of a simple- 

 harmonic force, since the acceleration of phase on one side of 

 is cancelled by the retardation on the other. 



Next suppose that the assisting impulses are given 

 each time the bob passes the symmetrically situated points 

 P, P' inwards. There is an acceleration of phase at each 

 impulse, and the period is shortened. This illustrates the case 

 of a disturbing force whose period is less than the natural 

 period, and whose maxima and minima precede the maxima and 

 minima of the velocity. If on the other hand the impulses are 

 given as the bob passes the points P and P' outwards, there is 

 a repeated retardation of phase, and the period is lengthened. 

 This corresponds to the case of a disturbing force whose period 



