CHAPTER IV.

PROBLEM OF DIFFERENT SOUNDS OF THE SAME

STRING.

IT had been observed at an early period of acoustical knowledge, that one string might give several sounds. Mersenne and others had noticed' that when a string vibrates, one which is in unison with it vibrates without being touched. He was also aware that this was true if the second string was an octave or a twelfth below the first. This was observed as a new fact in England in 1674, and communicated to the Royal Society by Wallis. But the later observers ascertained further, that the longer string divides itself into two, or into three equal parts, separated by nodes or points of rest; this they proved by hanging bits of paper on different parts of the string. The discovery so modified was again made by Sauveur about 1700. The sounds thus produced in one string by the vibration of another, have been termed Sympathetic Sounds. Similar sounds are often produced by performers on stringed instruments, by touching the string at one of its aliquot divisions, and are then called the Acute Harmonics. Such facts were not difficult to explain on Taylor's view of the mechanical condition of the

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string; but the difficulty was increased when it was noticed that a sounding body could produce these different notes at the same time. Mersenne had remarked this, and the fact was more distinctly observed and pursued by Sauveur. The notes thus produced

note of the string, have

in addition to the genuine verting been called secondary notes; those usually heard are, the octave, the twelfth, and the seventeenth above the note itself. To supply a mode of conceiving distinctly, and explaining mechanically, vibrations which should allow of such an effect, was therefore a requisite step in acoustics.

This task was performed by Daniel Bernoulli in a memoir published in 1755*. He there stated and proved the principle of the coexistence of small vibrations; it was already established, that a string might vibrate either in a single swelling (if we use this word to express the curve between two nodes which Bernoulli calls a ventre,) or in two or three or any number of equal swellings with immoveable nodes between. Daniel Bernoulli showed further, that these nodes might be combined, each taking place as if it were the only one. This appears sufficient to explain the coexistence of the harmonic sounds just noticed. D'Alembert, indeed, in the article Fundamental in the French Encyclopédie, and Lagrange in his Dissertation on Sound in the Turin Memoirs, offer several objections to this explanation; and it cannot be denied that the subject has

Berlin Mem. 1753, p. 147.

5 T. i, p. 64, 103.

its difficulties; but still these do not deprive Bernoulli of the merit of having pointed out the principle of coexistent vibrations, or divest that principle of its value in physical science.

Daniel Bernoulli's Memoir, of which we speak, was published at a period when the clouds which involve the general analytical treatment of the problem of vibrating strings, were thickening about Euler and D'Alembert, and darkening into a controversial hue; and as Bernoulli ventured to interpose his view, as a solution of these difficulties, which, in a mathematical sense, it is not, we can hardly be surprised that he met with a rebuff. The further prosecution of the different modes of vibration of the same body need not be here considered.

The sounds which are called Grave Harmonics, have no analogy with the acute harmonics abovementioned; nor do they belong to this section; for in the case of grave harmonics, we have one sound from the co-operation of two strings, instead of several sounds from one string. These harmonics are, in fact, connected with beats, of which we have already spoken; the beats becoming so close as to produce a note of definite musical quality. The discovery of the grave harmonics is usually ascribed to Tartini, who mentions them in 1754; but they are first noticed in the work of Sorge On tuning Organs, 1744. He there expresses this discovery in a query. "Whence comes it, that if we tune a fifth 2:3, a

• Chladni. Acoust. p. 254.

third sound is faintly heard, the octave below the lower of the two notes? Nature shows that with 2:3, she still requires the unity to perfect the order 1, 2, 3." The truth is, that these numbers express the frequency of the vibrations, and thus there will be coincidences of the notes 2 and 3, which are of the frequency 1, and consequently give the octave below the sound 2. This is the explanation given by Lagrange', and is indeed obvious.

7 Mem. Tur. i. P. 104.

CHAPTER V.

PROBLEM OF THE SOUNDS OF PIPES.

It was taken for granted by those who reasoned on sounds, that the sounds of flutes, organ-pipes, and wind-instruments in general, consisted in vibrations of some kind; but to determine the nature and laws of these vibrations, and to reconcile them with mechanical principles, was far from easy. The leading facts which had been noticed were, that the note of a pipe was proportional to its length, and that a flute and similar instruments might be made to produce some of the acute harmonics, as well as the genuine note. It had further been noticed', that pipes closed at the end, instead of giving the series of harmonics 1,,,, &c., would give only those notes which answer to the odd numbers 1,,, &c. In this problem also, Newton' made the first step to the solution. At the end of the propositions respecting the velocity of sound, of which we have spoken, he noticed that it appeared by taking Mersenne's or Sauveur's determination of the number of vibrations corresponding to a given note, that the pulse of air runs over twice the length of the pipe in the time of each vibration. He does not follow out this observation, but it obviously points to the

'D. Bernoulli, Berlin Mem. 1753, p. 150.

VOL. II.

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• Princip. Schol. Prop. 50.

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