32 DYNAMICAL THEORY OF SOUND 



13. Effect of Damping on Resonance. 



The abnormal amplitude and dissipation which ensue 

 whenever the imposed period is equal, or nearly equal, to the 

 natural period constitute the phenomenon of "resonance," 

 already referred to in 8, of which we shall have many 

 acoustical examples in the sequel. It may be illustrated 

 mechanically by giving a slight to-and-fro motion of suitable 

 period to the point of suspension of a simple pendulum, or 

 better by means of a double pendulum ( 14), i.e. an arrange- 

 ment in which two weights are attached at different points to 

 a string hanging vertically from a fixed point. If the upper 

 weight (M ) be considerable, whilst the lower one (ra) is relatively 

 small, M will swing almost exactly like the bob of a simple 

 pendulum, the reaction of ra being slight. Under these 

 conditions the motion of ra is practically that of a pendulum 

 whose point of suspension has an imposed simple-harmonic 

 vibration ( 8), and if the length of the lower portion of the 

 string be properly adjusted, a violent motion of ra may ensue. 



One very important point remains to be mentioned. As the 

 interval p/n between the forced and the natural frequencies 

 diverges from unity (on either side), the dissipation falls off 

 from its maximum the more rapidly, the smaller the value of 

 the frictional coefficient b. In other words, the greater the 

 intensity of the resonance in the case of exact coincidence of 

 frequencies, the narrower the range over which it is approxi- 

 mately equal to the maximum. For example, a tuning fork, even 

 when mounted on a "resonance box," requires very perfect tuning 

 in order that it may be excited perceptibly by the vibrations of 

 another fork in the neighbourhood, whereas the column of air 

 in a nearly closed vessel (e.g. a bottle or an organ pipe) will 

 respond vigorously to a much wider range of frequencies. To 

 elucidate the point, we notice that the expression (20) of 12 

 for the dissipation may be written 



,, 

 26 S1 



where = %nb/c = I/WT, ........ . ............ (2) 



