We have constant occasion in mechanical work to notice that whereas one pull or push of great vigour will not create some desired displacement of an object, a number of very small hits, or properly timed pushes or pulls, will achieve the requisite result. We might summarize the foregoing facts by saying that it is a maxim in dealing with bodies capable of any kind of free vibration that impulses, however small, will create oscillations of any required magnitude, if only applied at intervals equal to the natural free period of vibration of the body in question.
We can illustrate these principles by a few experiments which have special reference to musical instruments. If we fasten one end of a rope to a fixed support, we find we can produce a wave or pulse in the rope by jerking the free end up and down with the hand. The speed with which a pulse or wave travels along a rope depends upon its weight per unit of length, or, say, on the number of pounds it weighs per yard, and on the tension or pull on the rope. The tighter the rope, the quicker it travels; and for the same tension the heavier the rope, the slower it travels.
It is not difficult to show that the speed with which the pulse travels is measured by the square root of the quotient of the tension of the rope by its weight per unit of length, or, as it may be called, the density of the rope.
We have already explained that, in a medium such as air, a wave of compression is propagated at a speed which is measured by the square root of the quotient of the air-pressure, or elasticity, by its density. In exactly the same way the hump that is formed on a rope by giving one end of it a jerk, runs along at a speed which is measured by the square root of the quotient of the stretching force, or tension, by the density. The propagation of a pulse or wave along a string is most easily shown for lecture purposes by filling a long indiarubber tube with sand, and then hanging it up by one end. The tube so loaded has a large weight per unit of length, and accordingly, if we give one end a jerk a hump is created which travels along rather slowly, and of which the movement can easily be watched. We may sometimes see a canal-boat driver give a jerk of this kind to the end of his horse-rope, to make it clear some obstacle such as a post or bush.
If we do this with a rope fixed at one end, we shall notice that when the hump reaches the end it is reflected and returns upon itself. If we represent by the letter l the length of the rope, and by t the time required to travel the double distance there and back from the free end, then the quotient of 2l by t is obviously the velocity of the wave. But we have stated that this velocity is equal to the square root of the tension of the rope (call it e) by the weight per unit of length, say m. Hence clearly—
2l/t = √e/m ; or t = 2l · √m/e
Supposing, then, that the jerks of the free end are given at intervals of time equal to t, or to the time required for the pulse to run along and back again, we shall find the rope thrown into so-called stationary waves. If, however, the jerks come twice as quickly, then the rope can accommodate itself to them by dividing itself into two sections, each of which is in separate vibration; and similarly it can divide itself into three, four, five, or six, or more sections in stationary vibration. The rope, therefore, has not only one, but many natural free periods of vibration, and it can adapt itself to many different frequencies of jerking, provided these are integer multiples of its fundamental frequency.
The above statements may be very easily verified by the use of a large tuning-fork and a string. Let a light cord or silk string be attached to one prong of a large tuning-fork which is maintained in motion electrically as presently to be explained. The other end of the cord passes over a pulley, and has a little weight attached to it. Let the tuning-fork be set in vibration, and various weights attached to the opposite end of the cord.
It is possible to find a weight which applies such a tension to the cord that its time of free vibration, as a whole, agrees with that of the fork. The cord is then thrown into stationary vibration. This is best seen by throwing the shadow of the cord upon a white screen, when it will appear as a grey spindle-shaped shadow. The central point A of the spindle is called a ventral point, or anti-node, and the stationary points N are called the nodes ([see Fig. 54]). Next let the tension of the string be reduced by removing some of the weight attached to the end. When the proper adjustment is made, the cord will vibrate in two segments, and have a node at the centre. Each segment vibrates in time with the tuning-fork, but the time of vibration of the whole cord is double that of the fork. Similarly, by adjusting the tension, we may make the cord vibrate in three, four, or more sections, constituting what are called the harmonics of the string.
The string, therefore, in any particular state as regards tension and length, has a fundamental period in which it vibrates as a whole, but it can also divide itself into sections, each of which makes two, three, four, or more times as many vibrations per second.