A child is seated on a swing. His feet do not reach the ground. In order to swing him, one can, of course, raise the swing high up and then let it go. But this would be rather difficult and also quite unnecessary; it is enough to gently push the swing in time with the oscillations, and in a short time the child will be really swinging!

In order to swing a body, it is necessary to act in time with the oscillations. In other words, it is necessary to make one’s pushes occur with the same period as that of the free oscillations of a body. In such cases one speaks of resonance.

Resonance, widespread in nature and technology, merits careful consideration.

You can observe a very amusing and peculiar occurrence of resonance if you construct the following apparatus. Extend a string horizontally and hang three pendulums on it (Figure 1), two short ones of identical length and a longer one. Now deflect and release one of the short pendulums. In a few seconds you will see how the other pendulum of the same length will gradually begin oscillating too. A few more seconds—and the second short pendulum will swing, so that it will no longer be possible to tell which of the two pendulums first began moving.

What is the reason for this? Pendulums of the same length have identical periods of free oscillations. The first pendulum swings the second. The oscillations are transmitted from one to the other through the string connecting them. True, but there is yet another pendulum, of different length, hanging on the string. And what will happen to it? Nothing will happen to it. The period of this pendulum is different, and a short pendulum will not be able to swing it. The third pendulum will be present at this interesting energy “transfusion” from one pendulum to another, taking no part in it.

Each of us often comes across mechanical resonance phenomena. Perhaps you simply did not pay any attention to them, even though resonance is sometimes very bothersome. A streetcar passed by your window, and the dishes in the sideboard began jingling. What is the matter? Oscillations of the ground were transmitted to the building and simultaneously to the floor of your room, so your sideboard and the dishes in it started to vibrate. The oscillation was propagated so far and through so many objects. This happened as a result of resonance. The external oscillations were in resonance with the free oscillations of the bodies. Almost any rattling which we hear in a room, a factory or a car occurs because of resonance.

Resonance, as, incidentally, many phenomena, can be both useful and harmful.

A machine is standing on a mounting. Its moving parts move rhythmically, with a definite period. Imagine that this period coincides with that of free oscillation of the mounting. What will happen? The mounting will be soon vibrating, which could result in a breakdown.

The following fact is known. A company of soldiers was marching in step across a bridge in St. Petersburg. The bridge collapsed. An investigation into this matter was begun. It seemed that there were no grounds for anxiety over the fate of the bridge or the people: how many times had crowds gathered on this bridge, had heavy vehicles weighing much more than a company of soldiers slowly crossed it!

But a bridge sags by an insignificant amount under the action of a heavy weight. An incomparably greater sagging occurs when a bridge swings. The resonance amplitude of an oscillation can be thousands of times greater than the displacement caused by a stationary load of the same weight. This is precisely what the investigation showed – the period of free oscillation of the bridge coincided with that of an ordinary marching step.

Therefore, when a military subunit crosses a bridge, a command is given to break step. If people’s movements are not coordinated, resonance will not set in and bridges will not swing. Incidentally, this tragedy is well remembered by engineers. In designing bridges, they try to make its period of free oscillation far from the period of a marching step.

Designers of mountings have similar problems. They try to make the mounting in such a way that its period of oscillation be as far as possible from that of the moving parts of a machine.