Page images
PDF
EPUB

mass

For mstanc. in east afsave the waters of a great river flowing constanty dors toward the sea while waves are roling won the very sam part of the stream and wilt the great elevation, winci makes the tide, & maling ran the sea perhaps with * Velocity of fifty miles at hom. The motor of such a wave, or elevaZOL, Þ distma from any spean,, ale & d the late of undulations in genera.. The parts of the fini sir kr & stort time and for a small distance, so as to accumulate thenes al & eghboring part, and ther retire to ther former plaet, she is my enem affects the parts in the order of ther places. Perlads i tu Yesut looks at a field of standing corn. v hor gusts di vite, ai svetling over in visible waves, ht wil. have his conception & this matte wäre, for he will see that here, where each, cat d' grail is alichool. My is stalk, there can be no permanent loca, motol, a thi stilstanez, bir an" & successive stooping and rising of the scharake stres, Bouncing hollows and waves, closer

and luxer strins at the crowdóu 237%.

Newton hai, moreover, le colis de the mechanical consequences which such condensations and richiedons a the elastic medium, air, would produce 1. the parts of the Me, ish- Immering known laws of the elasticity at an, he showe«, 11 & Vory remarkabit proposition, the law according to whick the pen, is a ar min vibrate. We may observe, that n. this solubok, as 11 that of the vibrating string already mentioned, s ruke was exlileted, socorang 1 which the particles migh, esciliate, but not the iw to what they must conform. It was proved that, by taking the modal a ciet particle to be perfectly similar to that of a pondulunk, the faroos, descaped, by contraction and expansion, wore precisely snob as the matar, "equired; but it was not shown that no other type a' escludor Wolne vrst to the same accordance of torce and madok. Newton's reasoning also gave a determination of the spool, a' pranuar at the puses: it appeared that sound ought to troval with, the velocity which a body would require by tulling took through, hail, the ke li q” a homogeneous atmosphere 2 “the height of a homogeneous atmosphere" being the height which the ait must have, 11, ardor te product, at the earth's surthen, the notus, atmosphere presura, sudnosing no diminution of density to take place it, sscending "hx hocht is som 29,000 feet; and honor it tallowed thn, the velocity was 968 fees. This velocity is really considerably los than that at sound; but st the

[ocr errors]

time of which

we speak, no accurate measure had been established; and Newton persuaded himself, by experiments made in the cloister of Trinity College, his residence, that his calculation was not far from the fact. When, afterwards, more exact experiments showed the velocity to be 1142 English feet, Newton attempted to explain the difference by various considerations, none of which were adequate to the purpose; -as, the dimensions of the solid particles of which the fluid air consists;-or the vapors which are mixed with it. Other writers offered other suggestions; but the true solution of the difficulty was reserved for a period considerably subsequent.

Newton's calculation of the motion of sound, though logically incomplete, was the great step in the solution of the problem; for mathematicians could not but presume that his result was not restricted to the hypothesis on which he had obtained it; and the extension of the solution required only mere ordinary talents. The logical defect of his solution was assailed, as might have been expected. Cranmer (professor at Geneva), in 1741, conceived that he was destroying the conclusiveness of Newton's reasoning, by showing that it applied equally to other modes of oscillation. This, indeed, contradicted the enunciation of the 48th Prop. of the Second Book of the Principia; but it confirmed and extended all the general results of the demonstration; for it left even the velocity of sound unaltered, and thus showed that the velocity did not depend mechanically on the type of the oscillation. But the satisfactory establishment of this physical generalization was to be supplied from the vast generalizations of analysis, which mathematicians were now becoming able to deal with. Accordingly this task was performed by the great master of analytical generalization, Lagrange, in 1759, when, at the age of twenty-three, he and two friends published the first volume of the Turin Memoirs. Euler, as his manner was, at once perceived the merit of the new solution, and pursued the subject on the views thus suggested. Various analytical improvements and extensions were introduced into the solu tion by the two great mathematicians; but none of these at all altered the formula by which the velocity of sound was expressed; and the discrepancy between calculation and observation, about one-sixth of the whole, which had perplexed Newton, remained still unaccounted for. The merit of satisfactorily explaining this discrepancy belongs to Laplace. He was the first to remark' that the common law of the

Méc. Cél. t. v. L. xii. p. 96.

changes of elasticity in the air, as dependent on its compression, cannot be applied to those rapid vibrations in which sound consists, since the sudden compression produces a degree of heat which additionally increases the elasticity. The ratio of this increase depended on the experiments by which the relation of heat and air is established. Laplace, in 1816, published the theorem on which the correction depends. On applying it, the calculated velocity of sound agreed very closely with the best antecedent experiments, and was confirmed by more exact ones instituted for that purpose.

This step completes the solution of the problem of the propagation of sound, as a mathematical induction, obtained from, and verified by, facts. Most of the discussions concerning points of analysis to which the investigations on this subject gave rise, as, for instance, the admissibility of discontinuous functions into the solutions of partial differential equations, belong to the history of pure mathematics. Those which really concern the physical theory of sound may be referred to the problem of the motion of air in tubes, to which we shall soon have to proceed; but we must first speak of another form which the problem of vibrating strings assumed.

It deserves to be noticed that the ultimate result of the study of the undulations of fluids seems to show that the comparison of the motion of air in the diffusion of sound with the motion of circular waves from a centre in water, which is mentioned at the beginning of this chapter, though pertinent in a certain way, is not exact. It appears by Mr. Scott's recent investigations concerning waves, that the circular waves are oscillating waves of the Second order, and are gregarious. The sound-wave seems rather to resemble the great solitary Wave of Translation of the First order, of which we have already spoken in Book vi. chapter vi.

9

IT

CHAPTER IV.

PROBLEM OF DIFFERENT SOUNDS OF THE SAME STRING.

T had been observed at an early period of acoustical knowledge, that one string might give several sounds. Mersenne and others

9

Ann. Phys. et Chim. t. iii. p. 288. Brit. Ass. Reports for 1844, p. 361.

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 Sauveur3 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 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 in addition to the genuine note of the string, have 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 its difficulties; but

1 Harm. lib. iv. Prop. 28 (1636).

4

• Berlin Mem. 1753, p. 147.

2 Ph. Tr. 1677, April.

T. i. pp. 64, 103.

3 A. P. 1701.

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 above-mentioned; 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 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.

IT

CHAPTER V.

PROBLEM OF THE SOUNDS OF PIPES.

TT was taken for granted by those who reasoned on sounds, that the sounds of flutes, organ-pipes, and wind-instruments in general, con

• Chladni. Acoust. p. 254.

Mem. Tur. i. p. 104.

« PreviousContinue »