CHAPTER VII.

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

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

Suppose that a particle of air is vibrating between the extreme positions A and B, under 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 vibrations constituting the second tone fall on the particle, it is just on the point of starting from A towards B, under the influence of those of the first. Two extreme cases are now possible, depending on the movement which the particle

would have executed, had it been affected by the later-impressed vibration alone. First, suppose that 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 complete opposition to each other, are also exactly equal in extent. In this case, the air-particle, being solicited with equal intensity in two diametrically opposite directions, remains at rest, the two systems of vibrations completely neutralizing each other's effect. In general, however, these systems, even when equal in extent of vibration, are neither in complete opposi

tion 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. We have only to lay down the simple wave-forms corresponding to the constituent vibrations, and superpose them as in § 72. The reader will have noticed that the differences of relative motion described on p. 149 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 respectively. 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 resulting 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 level-line, i. e. no effect whatever occurs.

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 with an external objective existence; not as consisting merely in a state of motion of certain air-particles, and therefore liable, on the application of an opposite system of equal 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 notes 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 tone completely goes out. These positions are mid-way between the four in which the faces of the prongs are held flat before the ear. As the fork revolves from one of these positions of loud tone 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 its 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