August i, 191 8] 



NATURE 



437 



trated by three pendulums, all hanging from the same 

 tightly stretched horizontal cord. One pendulum had 

 a heavy bob, and by its swings moved the stretched 

 cord. It thus acted as driver, and applied forces to 

 the other two pendulums, which had light bobs, and 

 so were easily driven. Of these pendulums one was 

 shorter and one longer than the driver. They soon 

 settled to opposite phases after the heavy bob was set 

 in motion. Resonance curves showing the varied 

 responses of such driven pendulums as the tuning is 

 altered were then thrown on the screen. 



In the cases just dealt with the light bob is set in 

 motion at the expense of energy taken from the heavy 

 one. But on account of the great disparity of the 

 bobs, this loss entailed no appreciable diminution in 

 the vibrations of the heavy bob or driver. • 



Consideration was next given to the case where 

 equal bobs hang from a tight cord. While both pen- 

 dulums are hanging at rest one bob is struck. Its 

 vibrations disturb the other pendulum and set it in 

 motion. But, obviously, while the driven pendulum 

 gains an amplitude -equal to that first possessed by 

 the driver, the driver itself would have lost all its 

 motion. The other then becomes the driver in turn, 

 and transfers its energy back to what was originally 

 the driver. 



This palpable surging of the energy to and fro 

 between the two pendulums marks them as showing 

 what may be called coupled vibrations. In both 

 cases the action of the driver on the driven is recog- 

 nised. But in the case of coupled vibrations the 

 reaction of the driven on the driver is palpable and 

 recognised also, whereas in what are called forced 

 vibrations this reaction is undiscernible or ignored. 



In the case of coupled vibrations just shown the 

 vibrations of each pendulum seem quite simple, but 

 slowly and alternately wax and wane in amplitude — 

 that is, they exhibit what are termed " beats.'" But it 

 is well known that beats may be heard when two 

 musical tones of slightly differing pitch are sounded 

 together. Further, the number of beats per second is 

 thf difference of the frequencies of the two tones. 

 Thus the waxing and waning vibrations of either 

 pendulum may be regarded as the superposition of two 

 simple vibrations of slightly different periods. 



The next case studied was that of two precisely 

 similar pendulums connected by hanging one from the 

 bob of the other. One bob being started by a blow, it 

 appeared to execute simple vibrations. The other 

 moved with a pause or twitch instead of in simple 

 fashion. Further, neither pendulum showed the 

 waxing and waning of amplitude which was so marked 

 in the other case where both hung from a stretched 

 cord. 



The questions which now naturally arise are :■ — 

 (a) Why this contrast? and (b) Can the gap be 

 bridged? The solution is simple. The difference in 

 appearance is only a matter of different ratios of 

 periods of the superposed vibrations, and this, again, 

 is due to different values of tho couplinf^. to borrow 

 a term from electrical theory. We have changed sud- 

 denly from a very loose to a very tight coupling. We 

 consequently passed at a bound from periods nearly 

 equal (giving a slow- waxing and waning) to periods 

 the ratio of which exceeds 2 : i (involving the pause or 

 twitch) ; for the theorv shows that as the coupling 

 increases the ratio of the periods increases also. 



It is accordingly of interest to change the coupling 

 gradually and so bridge the gap between the two 

 motions which seemed so unlike. This was done bv 

 the cord-and-lath pendulum, in which the cord pen- 

 dulum is suspended from an adjustable stud on the 

 lath pendulum. When the two suspensions are near 

 together, the value of the coupling is almost equal to 



NO. 2544, VOL. lOl] 



the fraction of the lath-length at which the cord is 

 attached. When this fraction is unity, as in the case 

 of one pendulum hanging from the bob of the other, 

 the coupling has the value i/v'3, or 58 per cent.* nearly. 

 (These simple relations are for equal bobs and equal 

 pendulum lengths.) 



III. — Electrical Vibrations, Forced and Coupled. 



On passing along the second line of Table I. it was 

 noted how the various types of electric vibrations may 

 be obtained and the striking analogy to them pre- 

 sented by the mechanical cases already considered. 



Any electrical circuit containing a capacity and an 

 inductance may exhibit electrical vibrations. For the. 

 fundamental electrical conditions are there present, just 

 as the mechanical ones were in the case of a simple pen- 

 dulum. If the condenser is charged by a suitable means, 

 the quantity of electricity so displaced is urged to flow- 

 back again round the circuit by the electromotive force 

 of the charged condenser. If the resistance of the 

 circuit is small enough, the electromagnetic inertia 

 (measured by the inductance) ensures that the current 

 shall still flow after the condenser is discharged. 

 Thus its charge is reversed. So the vibrations con- 

 tinue until the energy is dissipated by the resistance 

 of the circuit. These are free electrical vibrations. 



As an example of forced electrical vibrations we may 

 think of a circuit with capacity and small inductance 

 (like that of a Fleming cymometer) placed not too 

 near to a circuit of similar frequency, but with much 

 greater inductance. Then the cymometer will respond 

 to the vibrations of the other — i.e. it will execute 

 forced vibrations. These will not appreciably diminish 

 the vibrations of the main circuit. 



But let two electrical vibration circuits of comparable 

 inductances and periods be placed together and started, 

 then there is not only the action of the driver, but also 

 a distinct reaction of the driven on the driver. Hence, 

 as the vibrations of one circuit start those of the other, 

 the latter by their growth check the former, causing 

 them to die away. Thus there may be an interchange 

 of energy between them. This, as we have seen With 

 pendulums, corresponds with the superposition of vibra- 

 tions of slightly differing periods, provided the action 

 and reaction are small and the interchange slow. 

 Further, it is known that if two such circuits are 

 closely coupled, these two periods differ more widely. 

 Hence a third circuit (say a cymometer) responding to 

 either of them may detect these separate periods by 

 giving a resonance curve with two humps instead of 

 one. 



IV. — Traces from Coupled Pendulums. 

 It has been seen that there is a certain general 

 analogy between mechanical and electrical vibrations, 

 whether free, ^forced, or coupled. The question now 

 arises as to whether this analogy may reach or ap- 

 proach a quantitative exactness in all or any respects, 

 and whether it can be utilised in any way. 



Various mechanical vibrating systems differ widely. 

 Some resemble the electrical case very closely, but none 

 appears to be completely and exactly analogous to them 

 in every detail. Indeed, the electrical case seems to 

 be slightly simpler than any mechanical analogy yet 

 put forward. But the differences are small, and the 

 mechanical analogy may be highly useful as affording 

 visible and tangible illustrations of those subtle elec- 

 trical vibrations which can be neither seen nor 

 handled. Especially is this the case if the model is 

 readily adjustable to represent the various relations 

 of the constants concerned and can be used for any 

 initial conditions. Thus from such analogies some 

 benefit may accrue to the non-mathematical student. 

 But perhaps the highest advantage is realised only 

 bv those who combine the mathematical with the 



