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CHAPTER VII.

ON THE INTERFERENCE OF SOUND, AND ON 'BEATS.'

74. In § 71 we examined the principle on which the general problem of the composition of vibrations is solved. We now approach certain very important particular cases of that problem, which it will be worth while to solve both independently and as instances of the method repeatedly applied in § 72.

Suppose that a particle of air is vibrating between the extreme positions A and B while convey

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ing a sustained simple tone produced by a tuning-fork, or stopped flue-pipe. Now let a second instrument of the same kind be caused to emit a tone exactly in unison with the first. We will assume that, when the waves of the second tone reach the particle, it is just on the point of starting from A towards B. Two extreme cases are now possible, depending on the movement which the particle would have executed in virtue of the later-impressed vibration alone.

First, suppose that movement to be from A along the line AB, either through a greater or less distance than AB, back again to A, and so on. Here the separate effects of the two sets of vibrations will be added together, the particle will, therefore, perform vibrations of larger extent than it would under either component separately. Next, suppose that, under the second set of vibrations alone, the particle would move from A in the opposite direction to its former course, i.e. along BA produced, shown by a dotted line in the figure. In this case the separate effects are absolutely antagonistic; accordingly the joint result is that due to the difference of its components. The particle will, therefore, execute less extensive vibrations than it would have done under the more powerful of the two components acting alone.

The most striking result presents itself when the two systems of vibrations, besides being in opposition to each other, are also exactly equal in extent. In this case the air-particle, being solicited by equal forces in opposite directions, remains at rest, the two systems of vibrations completely neutralising each other's effects. In general, however, these systems, even when equal in extent of vibration, are neither in complete opposition nor in complete accordance, but in an intermediate attitude, so as only partially to counteract, or support, each other. These conclusions

admit of being exhibited in a more complete manner

by means of associated waves. down the simple wave-forms

We have only to lay

corresponding to the

constituent vibrations, and superpose them as in § 72. The reader will have noticed that the differences of relative motion described on the preceding page are merely phase-differences.

Fig.54.

(1)

(2)

(3)

In Fig. 54, (1), (2), (3), we have two waves of unequal amplitudes in complete accordance, complete antagonism and an intermediate condition, respec

tively. In Fig. 55, a case of equal and opposite waves is shown. In (1) Fig. 54, the resultant wave is the sum, and in (2) the difference of the component waves. In (3), we get a wave of intermediate amplitude. These three resultant waves are necessarily simple, as otherwise two simple tones in unison would give rise to a composite sound, which would be absurd. In Fig. 55 the wave-form degenerates into the initial line, i.e. no effect whatever is produced.

Fig.55.

75. Thus, when one simple tone is being heard, we by no means necessarily obtain an increase of loudness by exciting a second simple tone of the same pitch. On the contrary, we may thus weaken the original sound, or even extinguish it entirely. When this occurs we have an instance of a phenomenon which goes by the name of Interference. That two sounds should produce absolute silence seems, at first sight, as absurd as that two loaves should be equivalent to no bread. This is, however, only because we are accustomed to think of Sound as

something possessing an external substantial existence; not as consisting merely in a state of motion of certain air-particles, and therefore liable, on the application of suitable forces, to be absolutely annihilated.

A single tuning-fork presents an example of this very important phenomenon. Each prong sets up vibrations corresponding to a simple tone, and the two tones so produced are of the same pitch and intensity. If the fork, after being struck, is held between the finger and thumb, and made to revolve slowly about its own axis, four positions of the fork with reference to the ear will be found

where the sound goes completely out. These positions are mid-way between the four in which the plane sides of the prongs are held straight before the ear. As the fork revolves from one of these positions of loud sound to that at right-angles to it, the sound gradually wanes, is extinguished in passing the Interference-position, reappears very feebly immediately afterwards, and then continues to gain strength until the quarter of a revolution has been completed.

76. The case of coexistent unisons has now been adequately examined: we proceed to enquire what happens when two simple tones differing slightly in pitch are simultaneously produced. The problem is,

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