Page:Elementary Text-book of Physics (Anthony, 1897).djvu/198

184 $$3n + m$$ vibrations, for instance, the first difference tone would make $$2n + m$$ vibrations. This tone and the one making $$n$$ vibrations would give the tone making $$n + m$$ vibrations; this tone, in turn, and the one making $$n$$ vibrations would give the tone making $$m$$ vibrations. This last tone is the one which is heard most plainly, and it seems difficult to admit that it can be the resultant of tones which are only heard very feebly, and often not at all. In Fig. 64 are represented the resultant curves produced in several of these cases. The first curve corresponds to two tones of which the vibration numbers are as 15:16. It shows the periodic increase and decrease in amplitude, occurring once every 15 vibrations, which, if not too frequent, give rise to beats (§ 139). If the pitch of the primaries be raised, preserving the relation 15:16, the beats become more frequent, and finally a distinct tone is heard, the vibration number of which corresponds to the number of beats that should exist. It was for a long time considered that the resultant tone was merely the rapid recurrence of beats. Helmholtz has shown by a mathematical investigation that a distinct wave making m vibrations will result from the coexistence of two waves making $$n$$ and $$n + m$$ vibrations, and he believes that mere alternations of intensity, such as constitute beats, occurring ever so rapidly cannot produce a tone.

In II and III (Fig. 64) are the curves resulting from two tones, the intervals between which are respectively 15:39(= 2 X 15 + 1) and 15: 31(= 3 X 15 + 1). Running through these may be seen a periodic change corresponding exactly in period to that shown in I. The same is true also of the curve in IV, which is the resultant for two tones the interval between which is 15:46(= 3 X 15 + 1). In all these cases, as has been already said (§ 155), if the pitch of the components be not too high, one beat is heard for every 15 vibrations of the lower component. Fig. 63 shows the flame images for the intervals $$n:n + m$$ and $$n:2n + m$$. The varying amplitudes resulting in $$m$$ beats per second are very evident in both. In all these cases, also, as the pitch of the components rises the beats become more frequent, and finally a resultant tone is heard, having, as already stated, one vibration for every 15 vibrations of the lower