it resounds to the fundamental it should not resound to the upper partials. We have seen that in the case of a closed cylindrical pipe the frequencies of the upper partials are 3, 5, 7, &c., times the frequency of the fundamental, so that if the sounding body, say a fork, has partials of any of these frequencies, these partials will be strengthened. In the case of resonators of the shape shown in Fig. 274, the frequency of the overtones is very high compared to the frequency of the fundamental, and so, if, as is the case with a fork, the upper partials which are present in any strength are comparatively low, the resonator will not be able to strengthen them. Thus for producing a pure tone the resonators of the form shown in Fig. 274, rather than simple closed pipes, are to be preferred. The periods of the fundamental tones of such resonators cannot usually be calculated, but the following general considerations apply: For a given opening or mouthpiece the pitch is mainly dependent on the volume of the enclosed air, while in the case of resonators without necks the influence of the mouth depends on its area.

314. Kundt's Experiment. If a rod AB (Fig. 271) is clamped at its middle so that one end projects into a tube CD, the end of the rod being fitted with a light piston which fits loosely into the tube, then, on causing the rod to vibrate longitudinally, this piston will vibrate backwards and forwards and will set up vibrations in the air contained in the tube CD.

A

B

D

C

FIG. 271.

The waves in the air in the tube will be reflected from the end D of the tube, and the direct and reflected waves will set up stationary vibrations in the air. If we suppose that the tube is closed at D, this point will be a node, and there will be a series of nodes along the tube at distances equal to /2 from one another, where A is the wave-length, in the gas which fills the tube, of the tone having a frequency equal to that of the rod. If the position of the end of the rod is adjusted so that the piston B is at a loop of these stationary vibrations, the motion of the piston will have its maximum effect in increasing their amplitude, and they will be so intense that if a light powder, such as lycopodium or cork filings, be strewn inside the tube, it will, by the vibration of the air or other gas, be collected in very characteristic transverse ridges at the loops. The explanation of the formation of these ridges is beyond the scope of this work, so we must content ourselves with referring the reader who wishes to pursue the subject to Rayleigh's "Sound," vol. ii. p. 46. By measuring the distance between consecutive loops, we obtain the value of X/2 for the tone produced by the rod in the gas, and this represents the space traversed by a sound-wave in the gas during the time the rod makes

this will also be

is the velocity of

half a vibration. If n is the frequency of the rod, the frequency of the vibrations in the gas, so that if sound in the gas, we have v=nλ, or v=2nl where is the distance between two of the loops in the tube. If the rod is giving its fundamental, then the wave-length of the sound in the rod is (§ 304) equal to 21, where L is the length of the rod. Hence for the material of which the rod is composed we have V=2nL, while for the gas in the tube v=2nl. Therefore

V]v=L]L.

Thus, by measuring the ratio of the length of the rod to the distance between two loops, we can calculate the ratio of the velocities of sound in the material of the rod and in the gas. If we know the frequency n of the rod, the velocity of sound in the gas can be calculated, so that by filling the tube with various gases we can obtain the velocity of sound in these gases. Without knowing n we can, by simply comparing the values of the wave-length, obtain the ratio of the velocities in the different gases. For if v and v' are the velocities of sound in two gases, and X and X' the wave-lengths corresponding to the tone of frequency n given by the rod, we have v=nλ, and v'=nλ', so that

where / and are the distances between consecutive loops as given by the Kundt's tube.

In the following table the value of the velocity of sound for some gases at a temperature of o° C. is given. (As has been pointed out in § 286, the velocity is independent of the pressure.)

CHAPTER VII

AUDITION, COMBINATION TONES, CONSONANCE,

AND VOCAL SOUNDS

315. Audition.-In considering the subject of the effects of sounds on the ear, we shall deal exclusively with the physical side of the subject, referring the reader to books on physiology for an account of that aspect of the subject.

The ear is not capable of detecting as sound the presence of airwaves of all frequencies, but it is only when the frequency of such waves falls between certain limits that the ear is able to distinguish their presence, and we experience the sensation we call sound. These limits are, however, neither of them well defined. Helmholtz concluded from his experiments that the lowest frequency which causes the sensation of a musical tone is about thirty vibrations per second. In forming any such estimate, it is very difficult to obtain a tone in which we may be quite certain no overtones are present; for, if they are present, what is actually heard may be the overtones and not the fundamental.

The upper limit of audibility is even more uncertain, for not only does it vary very much with the observer, but there is the added difficulty that it is very hard to determine the frequency of notes of very high pitch. The upper limit of audibility for normal ears appears to be somewhere between 10,000 and 20,000 vibrations per second. Estimates of pitch cannot, however, be made above a frequency of about 4000.

A closely related subject is the amplitude of the sound-waves in air necessary for audition. As a result of some experiments on the distance to which a whistle could be heard when a measured power (§ 78) was employed in maintaining the sound, Rayleigh came to the conclusion that under favourable circumstances the ear is able to detect a sound, if the amplitude of the sound-wave exceeds 10 cm.1

8

The direction from which a sound comes can be judged with considerable accuracy, and although the exact method by which we are able to make this estimate of direction is not known, there is no doubt that we are very largely guided by the effect of the sound on the two ears; probably the slight difference of the intensity with which the sound reaches the two ears is at the base of all such judgments.

1 Intermittent sounds can be detected by the ear when a continuous sound of the same amplitude is inaudible.

316. Beats.-We have seen in § 54 that when we combine two S.H.M.'s of very nearly the same frequency, of which the displacements take place in the same direction, the resultant motion is periodic, and the amplitude of the motion undergoes periodic variations, caused by the displacement due to the two motions sometimes being in the same direction, and thus aiding each other, and at other times being in opposite directions, and so opposing each other. In the case of sound, we may obtain a similar result, for when two tones, whose frequencies do not differ by more than about sixteen vibrations per second, are sounded together, a periodic waxing and waning of the sound due to the two tones occurs. Under such circumstances the tones are said to beat. The production of beats may be illustrated objectively by an arrangement similar to that used to produce Lissajous' figures (§ 301). The two forks are arranged to vibrate in the same plane, so that the amplitude of the movement of the spot of light on the screen is equal to the sum of the amplitudes due to each fork separately. It will then be seen that at each beat the spot of light is drawn out into a line, while

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FIG. 272.

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half-way between the beats the spot appears round. By projecting the spot of light on a photographic plate, which is moved in a direction at right angles to the plane of vibration of the forks, a curve such as that given in Fig. 272 is obtained, in which the effect of the beats is very clearly shown.

Let one tone (4) make a vibrations, while the other (B) makes 1+1. If then we start when the two are in the same phase, the phase of the tone B will gain on that of A, till, at the end of a vibrations of 4, B will have made +1 vibrations, and so they will again be in the same phase, and the sound will be a maximum. Let the frequency of the tone A be nx, and hence that of B n(x+1). Now from one maximum of sound to the next makes a vibrations, so that the number of maxima in a second will be n, or there will be n beats per second. But the difference between the frequencies of A and B is n(x + 1) − nx or n, so that the number of beats per second is equal to the difference in the frequencies of the two tones.

Starting with two tones in unison, and increasing the frequency of one of them, beats will be produced which are at first slow, but increase

in frequency as the difference in the frequencies of the tones increases. After a time the frequency of the beats will be so great that the ear ceases to hear them as such, and the only sensation is one of discord. While the beats are still audible, the ear is unable to distinguish the separate tones which are producing the beats, but when the beats cease to be distinguishable, then the ear can detect the existence of the two separate tones.

Koenig has advanced the theory that in the case of the beats produced by pure tones, such as those given by massive tuning-forks when lightly bowed, there are really two sets of beats. Thus suppose we have a fork giving 64 vibrations per second, this, according to what we have said above, will give 8 beats per second with a fork of which the frequency is 72. According to Koenig it is, however, possible to obtain 8 beats per second, if the second fork has a frequency of 120. In the first case we have 72-64-8, and in the second case 64 × 2-120=8, so that beats occur not only where the frequencies of two notes are nearly the same, but also when the frequency of the higher note is nearly equal to the frequency of the octave of the fundamental. Of course the presence of this superior series of beats is explained, if we suppose that the octave of the lower tone is given by the tuning-fork as an overtone, so that the beats heard are really due to the combination of this overtone with the higher fork. Koenig, however, maintains that the tones he uses are pure, that is, that no overtones are present.

Koenig's two series of beats are related to the primary tones in the following manner. The frequency m of the higher tone must lie. between two multiples, a and b, say, of the frequency n of the lower tone, where an is less than m, and bn is greater than m. Then the two series of beats which may be produced are an inferior series, in which there are man beats per second, and a superior series, in which there are bn-m beats per second. Thus if the two tones had frequencies of 40 and 74, then a=1 and b=2, since 74 lies between 40×1 and 40 × 2.

Hence the possible beats will be an inferior beat, of which the frequency is 74-40 or 34, and a superior beat, of which the frequency is 80-74 or 6. Both sets of beats are not, however, usually

audible at the same time.

317. Combination Tones. There are other phenomena besides the beats or throbbing sensation, which are due to the simultaneous production of two tones. For under certain circumstances, when two tones are sounded simultaneously, the ear is able to detect, in addition to the two primary tones, other musical tones, which are due to the combined effect of the two primary tones. These additional tones are called combination tones. There are three kinds of combination tones, namely, difference tones, summation tones, and beat tones.

The difference tone is a combination tone the frequency of which is equal to the difference in the frequencies of the two primary tones.