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ACOUSTICS If, instead of considering one point in a succession of to our actual sensations. The combination tones thus proinstants, we consider a succession of points along the line of duced in the source should have a physical existence in the and the amplitudes of those represented in (1) should propagation at the same instant, we evidently have waves air, be of the same order. The conditions assumed in this invesof amplitude varying from 2a down to 0, and then up to tigation are probably nearly realized in a harmonium and in a 2a again in distance U/^ -n2). If the difference of double siren of the form used by Helmholtz (0. A. § 51), and in these cases there can be no doubt that actual objective tones are frequency of the two tones is so great that the beats are not heard separately, and if the two sounds are of sufficient produced, for they may be detected by the aid of resonators of the frequency of the tone sought for. If the tones had no existence loudness, then a tone is heard of the same frequency, outside the ear then resonators would not increase their loudness. n i ~ n2’ as ti16 beats. This tone was first discovered by There is not much difficulty in detecting the difference tone by Sorge in 1740, and independently a few years later by a resonator if it is held, say, close to the reeds of a harmonium, Tartini, after whom it is named. It may easily be heard and Helmholtz succeeded in detecting the summation tone by the of a resonator. Further, Rucker and Edser, using a siren as when a double whistle with notes of different pitch is aid source, have succeeded in making a fork of the appropriate pitch blown strongly, or when two gongs are loudly sounded respond to both difference and summation tones {Phil Maq close to the hearer. It is heard, too, when two notes on xxxix. 1895, p. 341). But there is no doubt that it is very the harmonium are loudly sounded. Formerly, it was difficult to detect the summation tone by the ear, and many v oikers ha e doubted the possibility, notwithstanding the evidence generally supposed that the Tartini tone was due to the of such an observer as Helmholtz. Probably the fact noted by beats themselves, that the mere variation in the amplitude Mayer {Phil. Mag. ii. 1878, p. 500, or Rayleigh, Sound, § 386) that was equivalent, as far as the ear is concerned, to a super- sounds of considerable intensity when heard by themselves are position on the two original tones of a smooth sine dis- liable to be completely obliterated by graver sounds of sufficient force goes far to explain this, for the summation tones are of placement of the same periodicity as that variation. This course always accompanied by such graver sounds. view has still some supporters, and among its recent (2) The second mode of production of combination tones by advocates are Koenig and Hermann. But it is very diffi- the mechanism of the receiver is discussed by Helmholtz {Sensacult to suppose that the same sensation would be aroused tions of Tone, App. XII.) and Rayleigh {Sound, § 68). It depends on the restoring force due to the displacement ot the by a truly periodic displacement represented by a smooth receiver not being accurately proportional to the displacement. curve, and a displacement in which the period is only in This want of proportionality will have a periodicity, that of the the amplitude of the to-and-fro motion, and which is impinging waves, and so will produce vibrations just as does the represented by a jagged curve. No explanation is given variation of pressure in the case last investigated. We may see by the supposition; it is merely a statement which can how this occurs by supposing that the restoring force of the receiving mechanism is represented by hardly be accepted unless all other explanations fail. x + gx2, Helmholtz has given a theory which certainly accounts 2 for the production of a tone of the frequency of the beats, where x is the displacement and gx is very small. Let an external force F act on the system, and for simplicity suppose its Combina ant^ ^°r °^er tones all grouped under the name period is sp great compared with that of the mechanism that we tiontoTes. of combination tones. The only question is may take it as practically in equilibrium with the restoring force. whether the intensity of the tones heard is Then F = Xa; + gx2. accounted for by the theory. Combination tones may be 2 produced in three ways : — (1) In the neighbourhood of Now gx is very small compared with x, so that x is nearly equal the source; (2) in the receiving mechanism of the ear; to F/X, and as an approximation, (3) in the medium conveying the waves. I’ = x + ^ (1) We may illustrate the first method by taking a case disF F2 cussed by Helmholtz {Sensations of Tone, App. xvi.) where the or two sources are reeds or pipes blown from the same wind-chest. Let us suppose that with constant excess of pressure, p, in the now that F = a sin 2x«/ + b sin 2ir7i2t, the second term will wind-chest, the amplitude produced is proportional to the pressure, Suppose a series of combination tones of periodicities so that the two tones issuing may be represented by pa sin 27m1< evidentlyn produce 2>h> 2«2> i ~ n2’ an(I ni + n2, as in the first method. There can be and pb sin 2wnf. Now as each source lets out the wind periodi- no doubt that the ear is an unsymmetrical vibrator, and that it cally it affects the pressure in the chest so that we cannot re- makes combination tones, in some such way as is here indicated, gard this as constant, but may take it as better represented by out of two tones. Probably in most cases the combination p + a sin [f.Tnxf + e) + sin {2irn2t +f). Then the issuing dis- tones whichpure we hear are thus made, and possibly, too, the tones turbance will be detected by Koenig, and by him named “ beat-tones.” He found P + ^ sin {Zn-nf + e)+pb sin {2irn2t +/)} {a sin 2?™^ + b sin 2Tin.,t} that if two tones of frequencies p and q are sounded, and if q =pa sin 2Trn1t +pb sin 2Trn2t lies between Np and (N + l)p, then a tone of frequency either (N + ljp-?, or of frequency q - Np, is heard. The difficulty in a2 «2 . . + -y— Helmholtz’s theory is to account for the audibility of such beat A cos e ——A cos {‘iirnf + e) tones when they are of a higher order than the first. Rucker and Edser quite failed to detect their external existence, so that + 1~ cos/cos (Ittji/ +/) apparently they are not produced in the source. If we are to assume that the tones received by the ear are pure and free (tb . cM from partials, the loudness of the beat-tones would appear to + cos {27r(?i1 - n<^t + e} —cos {2'jr{n1 -f n^)t -f e} show that Helmholtz’s theory is not a complete account. (3) The third mode of production of combination tones, the cos a + WT {^{nx - n2)t -/} - ~. cos (2x(n1 + nf)t +/} (1) production in the medium itself, follows from the varying velocity parts of the wave, as investigated at the beginning of Thus, accompanying the two original pure tones there are (1) the of different article. It is easily shown that after a time we shall have octave of each ; (2) a tone of frequency (»i - n2) ; (3) a tone of this to superpose on the original displacement a displacement proporfrequency (% + n-f. The second is termed by Helmholtz the differ- tional to the square of the particle velocity, and this will introence tone, and the third the summation tone. The amplitudes of duce just the same set of combination tones. But probably in these tones are proportional to the products of a and b multiplied practice there is not a sufficient interval between source and by X or p.. These combination tones will in turn react on the hearer these tones to grow into any importance, and they pressure and produce new combination tones with the original can at for be only a small addition to those formed in the tones, or with each other, and such tones may be termed of the source ormost the ear. second, third, etc. order. It is evident that we may have tones Authorities.—Lord Rayleigh, Sound, 2nd ed. London, 1894, of frequency is the standard treatise on the phenomena of Acoustics; the last hni hi2 hnx - Tcn2 hn1 + kn2, on “Facts and Theories of Audition,” is a most valuwhere h and k are any integers.2 But inasmuch as the successive chapter, summary of the present state of knowledge of audition and orders are proportional to X X X3, or n /v>, and X and /x are able the different theories held.—Helmholtz, Sensations of Tone, 2nd small, they are of rapidly decreasing importance, and it is Eng. ed. London, 1885, deals more with the physiological aspect. not certain that any beyond those in equation (1) correspond —Sedley Taylor, Sound and Music, London, 1873, is a general