Waves and ripples in water, air, and æther

by Sir J. A. Fleming

In scientific language this is called the isochronism of the pendulum, and is said to have been discovered by Galileo in the Cathedral at Pisa, when watching the swings of a chandelier die away, whilst counting their number by the beats of his pulse. This periodic time of vibration, which is independent of the amplitude of vibration, provided the latter is small, is called the natural time of vibration of the pendulum, or its free periodic time.

In the case of the simple pendulum the free periodic time is proportional to the square root of the length of the pendulum. Accordingly, a short pendulum makes more swings per minute than a long one, and this rate of swinging is quite independent of the weight of the bob. We can, of course, take hold of the bob with our hand and force it to vibrate in any period we please, and thus produce a forced vibration; but a free vibration, or one which is unforced, has a natural time-period of its own.

In order that any body may vibrate when displaced and then set free, two conditions must exist. In the first place, there must be a controlling force tending to make the substance return to its original position when displaced. In the second place, the thing moved must have mass or inertia, and when displaced and allowed to return it must in consequence overshoot the mark, and acquire a displacement in an opposite direction. In the case of the pendulum the elastic control or restoring force is the weight of the bob, which makes it always try to occupy the lowest position. We can, however, make a pendulum of another kind. Here, for instance, is a heavy ball suspended by a spiral spring ([see Fig. 53]). If I pull the ball down a little, and then let it go, it jumps up and down, and executes vertical vibrations. The elastic control here is the spring which resists extension. In this instance, also, there is a natural free time of vibration, independent of the extent of the motion, but dependent upon the weight of the ball and the stiffness of the spring.

Fig. 53.

A good illustration of the above principles may be found in the construction of a clock or a watch. A clock contains a pendulum which vibrates in a certain fixed time. The arrangements we call the “works” of a clock are only a contrivance for counting the swings, and recording them by the “hands” of the clock. Owing, however, to the friction of the “works,” the pendulum would soon come to rest, and hence we have a mainspring or “weights” which apply a little push to the pendulum at each swing, and keep it going. In a watch there is no pendulum, but there is a “balance-wheel and hair-spring,” or a wheel which has a spiral spring attached to it, so that it can swing backwards and forwards through a small angle. The so-called “escapement” is a means by which the swings are counted, and a little impulse given to the wheel to keep it swinging. The watch “keeps time” if this hair-spring is of the right degree of stiffness, and the balance-wheel of the right weight and size. Thus a clock can be made to go faster or slower by slightly altering the length of its pendulum, and the watch by slightly changing the stiffness of its hair-spring.

It may be noted in passing that our legs, in walking, swing like pendulums, and every particular length of leg has its own natural time of vibration, so that there is a certain speed at which each person can walk which causes him or her the least amount of fatigue, because it corresponds with the natural free or unforced period of vibration of the leg considered as a pendulum.

We now pass on to notice another very important matter. If we have any pendulum, or mass suspended by a spring, having therefore a certain natural period of vibration, we can set it in motion by administering to it small repeated blows or pushes. If the interval between these impulses corresponds with the natural time-period of oscillation, it will be found that quickly a very large swing is accumulated or produced. If, on the other hand, the interval between the blows does not correspond with the natural time of vibration, then their effect in producing vibration is comparatively small. This may be illustrated with great ease by means of the ball suspended by a spring. Suppose that by means of an indiarubber puff-ball I make a little puff of air against the suspended ball. The small impulse produces hardly any visible effect. Let this puff be repeated at intervals of time equal to that of the natural free period of vibration of the suspended ball. Then we find that, in the course of a very few puffs, we have caused a very considerable vibration or swing to take place in the heavy ball. If, however, the puffs of air come irregularly, they produce very little effect in setting the ball in motion. In the same manner a pendulum, consisting of a heavy block of wood, may be set swinging over a considerable range by a very few properly timed taps of the finger. We may notice another instance of the effect of accumulated impulses when walking over a plank laid across a ditch. If we tread in time with the natural vibration-period of the flexible plank, we shall find that very soon we produce oscillations of a dangerously large extent. Whereas, if we are careful to make the time of our steps or movement disagree with that of the plank, this will not be the case.

It is for this reason that soldiers crossing a suspension bridge are often made to break step, lest the steady tramp of armed men should happen to set up a perilous state of vibration in the bridge. It is not untruthful to say that a boy with a pea-shooter could in time break down Charing Cross Railway Bridge over the Thames. If we suppose a pea shot against one of the sections of this iron bridge, there is no doubt that it would produce an infinitesimal displacement of the bridge. Also there is no question that the bridge, being an elastic and heavy structure, has a natural free time of vibration. Hence, if pea after pea were shot at the same place at intervals of time exactly agreeing with the free time-period of vibration of the bridge, the effects would be cumulative, and would in time increase to an amount which would endanger the structure. Impracticable and undesirable as it might be to carry out the experiment, it is nevertheless certainly true, that a boy with a pea-shooter, given sufficient patience and sufficient peas, could in time break down an iron girder bridge by the accumulation of properly timed but infinitely small blows.

The author had an instance of this before him not long ago. He was at a place where very large masts were being erected. One of these masts, about 50 feet long, was resting on two great blocks of wood placed under each end. This mast was a fine beam of timber, square in section, and each side about 2 feet wide. The mast, therefore, lay like a bridge on its terminal supports. Standing or jumping on the middle of this great beam produced hardly any visible deflection. The writer, however, placed his hand on the centre of the log and pressed it gently. Repeating this pressure at intervals, discovery was soon made of the natural time-period of vibration, and by repeating the pressures at the right moment it was found that large oscillations could be accumulated. If he had ventured to proceed far with this operation, it is certain that, with properly timed impulses, it would have been possible, by merely applying the pressure of one hand, to break in half this great wooden mast.