§ 5. Vibrations of a Tuning-fork
From the vibrations of a bar free at both ends it is easy to pass to the vibrations of a tuning-fork, as analyzed by Chladni. Supposing a a, Fig. 62, to represent a straight steel bar, with the nodal points corresponding to its first mode of division marked by the transverse dots. Let the bar be bent to the form b b; the two nodal points still remain, but they have approached nearer to each other. The tone of the bent bar is also somewhat lower than that of the straight one. Passing through various stages of bending, c c, d d, we at length convert the bar into a tuning-fork e e, with parallel prongs; it still retains its two nodal points, which, however, are much closer together than when the bar was straight.
Fig. 62.
Fig. 63.
When such a fork sounds its deepest note, its free ends oscillate as in Fig. 63, where the prongs vibrate between the limits b and n, and f and m, and where p and q are the nodes. There is no division of a tuning-fork corresponding to the division of a straight bar by three nodes. In its second mode of division, which corresponds to the first overtone of the fork, we have a node on each prong, and two at the bottom. The principle of Young, referred to at page 155, extends also to tuning-forks. To free the fundamental tone from an overtone, you draw your bow across the fork at the place where the node is required to form the latter. In the third mode of division there are two nodes on each prong and one at the bottom; in the fourth division there are two nodes on each prong and two at the bottom; while in the fifth division there are three nodes on each prong and one at the bottom. The first overtone of the fork requires, according to Chladni, 6-1/4 times the number of vibrations of the fundamental tone.
It is easy to elicit the overtones of tuning-forks. Here, for example, is our old series, vibrating respectively 256, 320, 384, and 512 times in a second. In passing from the fundamental tone to the first overtone of each you notice that the interval is vastly greater than that between the fundamental tone and the first overtone of a stretched string. From the numbers just mentioned we pass at once to 1,600, 2,000, 2,400, and 3,200 vibrations a second. Chladni’s numbers, however, though approximately correct, are not always rigidly verified by experiment. A pair of forks, for example, may have their fundamental tones in perfect unison and their overtones discordant. Two such forks are now before you. When the fundamental tones of both are sounded, the unison is perfect; but when the first overtones of both are sounded, they are not in unison. You hear rapid “beats,” which grate upon the ear. By loading one of the forks with wax, the two overtones may be brought into unison; but now the fundamental tones produce loud beats when sounded together. This could not occur if the first overtone of each fork was produced by a number of vibrations exactly 6-1/4 times the rate of its fundamental. In a series of forks examined by Helmholtz, the number of vibrations of the first overtone varied from 5·6 to 6·6 times that of the fundamental.
Starting from the first overtone, and including it, the rates of vibration of the whole series of overtones are as the squares of the numbers 3, 5, 7, 9, etc. That is to say, in the time required by the first overtone to execute 9 vibrations, the second executes 25, the third 49, the fourth 81, and so on. Thus the overtones of the fork rise with far greater rapidity than those of a string. They also vanish more speedily, and hence adulterate to a less extent the fundamental tone by their admixture.