I6 Interference and beats
The wave form of the motion of sound explains certain sound phenomena as being the result of a superposition of two sound processes. Like on water two systems of waves under certain conditions reinforce or weaken and even cancel each other (Fig. 297), also two sound wave systems - one tone joining a second one - can provoke silence. The reaction of surprise vanishes when you disregard sound sensation and only think of two motions, which can strengthen or weaken each other. If the air in the auditory canal has received an oscillation, which corresponds to the curve a and a simultaneous second one b (Fig. 349), the ear hears a strengthened tone and the result of their combined action is the curve c. However, if the second tone b arrives with a phase shift of half a wave length with respect to a, that is, if it attempts to excite a vibration, which is at each instant opposite to the first (curve b in Fig. 350), then the combined action is represented by c' - a straight line - and the ear does not hear a sound.
This phenomenon is by no means common. In order for two equal waves to extinguish each other, they must arrive at the ear with an exactly opposite phase; this happens in the case of the various kinds of reflections, which a sound ann encounter ordinarily , but only accidentally and for a brief instant. However, you can reconstruct the two situations of Figs. 349/350, for example, with the experiment due to Rudolph König 1832-1901 (Fig. 351). Instead of two equal sources of sound at different locations you employ only one source - a tuning fork, the tone of which is amplified by a resonator - and conduct the tone to your ear along two different paths (tubes), that is, you decomposes effectively the source of sound into two sources. The length of one path - in the figure the left one - can be changed (like in a trombone), so that you can make both paths equally or differently long. If you make them equally long, the waves arrive at the ear along both paths with the same phase: You hear the tuning fork (case c in Fig. 349); similarly, by lengthening the variable path by 1, 2, ˇˇˇ , n complete wave lengths. However, if you lengthen the path by 1, 3, ˇˇˇ, n half wave lengths, you obtain the case of Fig. 350 c'; the waves reach your ear at each instant with opposite phase: You do not hear anything. (This can also be demonstrated with manometric flames and a rotating mirror: If the tones extinguish each other, you see a simple strip, otherwise spikes.)
A tuning fork yields a surprising interference phenomenon (Wilhelm Eduard Weber), which is due to two spikes moving simultaneously towards and away from each other. If you rotate about its longitudinal axis a struck tuning fork, held vertically in front of your ear,, you will hear it clearly in four definite positions: Namely, when during the rotation of the fork the line ab or the line df lie in the direction of the auditory canal. In between these four positions, you will not hear it at all in four positions: Namely, when the lines ki or gh are in those direction. If you positions the tuning fork in one of the latter positions and push a small tube over one of its prongs without disturbing its oscillation, you will again hear the sound, because then the effect of the other prong reaches the ear undisturbed.
A remarkable interference phenomenon is obtained from two equally highly covered lip pipes (organ pipes) employing the same blower: They extinguish one another almost completely. and adjust to each other so that while the air enters the one and it leaves the other, whence you ear receives two waves with opposite phase. This can be demonstrated by means of a rotating mirror. (Two open lip pipes or two tongue pipes of equal construction and with equal tuning behave under these conditions differently, because they have overtones. They switch to the higher octave.)
If the two interfering tones are not perfectly, but almost equally high (their periods are almost the same), also their wave lengths differ very little - in the higher tone, two bulges follow each other at slightly shorter distances than in the deeper one, then neither the case of Fig. 349c nor that of Fig. 350c' will occur. A new phenomenon takes place: You hear a tone which intermittently strengthens and weakens: Beats. Fig. 353 (a partner to Figs. 349/350) shows the formation of the strengthening and weakening: a and b represent two simple tones (without overtones), a and b execute 27 and 30 oscillations, respectively: c represents the resulting oscillation, in which the amplitude grows gradually by superposition of both oscillations, then decreases again, etc. Increase and decrease of the amplitude implies increase and decrease of the strength of the tone.
The interference tone is loudest when two wave bulges meet. How often does this happen during each second? At the instant, when both tones start, their bulges coincide and the tone is loudest. The higher tone hastens ahead of the lower one; if it executes (n + 1) oscillations while the other executes n, this means that the (n + 1)th bulge of the higher tone coincides with the nth bulge of the deeper one. The interference tone is again strong. That is, you hear a beat as often as the higher tone is just ahead of a bulge of the lower one. (Exactly in between two beats, of course, only for a brief instant, the tone disappears altogether, since a bulge of the one tone coincides with the valley of the other tone.) If the lower tone has r, the higher tone s Hertz (oscillations per sec), how many beats will you then hear during one second? Assuming that you hear during 1 second x beats, you hear each 1/x second one beat, that is, after every 1/x second the difference between the oscillations, executed by the two tones, is 1. However, in 1/x sec, the deeper tone executes r/x, the higher tone s/x oscillations, whence s/x - r/x = 1, that is, the number of beats in one second equals the difference of the Hertz numbers of the interfering tones. You can demonstrate this by means of the double siren of Helmholtz (essentially a combination of two sirens of Dove 1803-1879).
You can produce beats with all instruments, especially a clear one with those which do not have any or only weak overtones - tuning forks and covered pipes; for example, with two tuning forks which give the same tone already by the one being slightly out of tune, say, by sticking a small piece of wax to it. In the case of such instruments, the interference tone disappears at the centre between two maxima (they are called pulses). That is the instant at which a bulge and a valley coincide. The strengthening and weakening of the combined tone then becomes very clear. - While in the case of instruments with loud overtones the fundamental tone vanishes between two pulses, the first overtone prevails so strongly that the tone switches to the octave. - You can also demonstrate beats as a proof that they do not depend on the ear; Fig. 353 shows a record of a phonoautograph. - You employ beats as auxiliary tools for tuning a tuning fork or a string of a piano; a presence or absence of beats between a normal tuning fork and the fork to be tuned (or of the piano string) indicates whether the normal tone height has been achieved or not.
A technical application of great importance is the intended generation of beats in the telephone receiver of telephony. The border of audibility of tones lies around 20 - 50 thousand Hertz, while the Hertz number of the modulated oscillations coming from the antenna is of the order of several hundred thousands; already in the case of an electric wave of length 1 km, it is 300000. In order to separate the tone, arriving in the form of electric vibrations, and to give it an audible Hertz number, you send through the receiver simultaneously a second oscillation (auxiliary oscillation), generated by a special generator, for example, with 301000 Hertz. The two oscillations then interfere, yield 1000 beats per second and thereby present an audible tone (beat reception).
Consonance and dissonance
Our sensing of the consonance of two tones (which have no overtones) depends essentially on how many beats per second are formed by the two tones. Beats give consonance a buzzing effect. - Helmholtz says a certain roughness - at a certain number a certain rattling. This does not disturb the consonance significantly as long as the sounds form at the most 4 - 6 beats per second. The perturbation grows with the number of beats; according to Helmholtz it reaches its maximum at 33 and then decreases and vanishes at 132. That many beats per second cannot be counted, but they provoke at this frequency a certain sensation. (The rough, rattling tone in the ear can be compared with the impression made to the eye by a flickering candle. We can also not count the flickering in such lights, since we sense this phenomenon quite differently from enumerable consecutive quivers.) If the degree of consonance really depends on the number of beats, then the same interval must have a different degree of consonance in the high section of the scale than in the deep one, because it causes there more beats than in the lower one. And indeed: For example, the interval h1 - c2 with 489 - 522 = 33 beats sounds like a cutting dissonance, the same interval H - c with 61 - 65 = 4 beats sounds much less harsh. On the other hand, the same interval can be at the high level a consonance, at the low level a dissonance - in fact, at the high level, the number of beats may amount to more than 133 per second, whence they are not perceptible, at the low level, their number may lie in the range of audibility. For example, this is the case with the large Third: At the high level, it is a consonance, at the low level, it approaches a not disregardable dissonance.
However. the number of beats is not the only cause for roughness of consonance; otherwise, for example, the intervals with the same number of beats (33)
|G 97-c 130||c130 - e 163||e 163 - g 196||c1 261 - d1 294||h1 489 - c2 522|
should have the same roughness. However, the deeper intervals, the larger ones, sound less rough, which means: Also the size of the interval influences the roughness of consonance. Helmholtz explains that one and the same Corti arc or one and the same nerve fibre reacts simultaneously to two different tones, if they lie as interval sufficiently close together, in order to be able to excite the same fibre of the membrana basilaris - otherwise the tone, which is too far away, resonates only so weakly, that the beats become too weak to be perceived.
If two different high tones continue to sound simultaneously very strongly and regularly, there arise (at a suitable interval and strength of these tones) combination tones: The difference-tones (discovered 1740 by the Hamburg organist Georg Andreas Sorge 1703-1778 are also called Tartini tones after Giuseppe Tartini 1692-1770 who discovered them independently soon after) have vibration numbers equal to the difference of the oscillation numbers of those two tones and the summation-tones equal to the corresponding sum. Especially the difference-tone can become very strong. According to Helmholtz's mathematical theory, oscillations of air, the number of which corresponds to the height of the combination-tones, must arise whenever the two primary tones excite an elastic body simultaneously so strongly that its oscillations can no longer be considered to be infinitely small. According to Rudolf König, the existence of combination tones contradicts the correctness of the resonance theory of hearing. Helmholtz explained their formation by deviations of the principle of undisturbed superpositions of primary tone oscillations; according to his view, they do not arise outside the ear, but only in the ear drum. However, Waetzmann has also generated outside the ear very strong combination tones (with one amplitude, which is a multiple of the primary tones) and has brought the theory by an extension of Helmholtz's concepts and a compromise between Helmholtz and König to a certain conclusion.
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