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from without, but is forced by its own pressure to rush along an opened pipe. In this case the amount of energy of the mass of gas remains unaltered. If, then, the gas

were under the pressure

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when at rest, and its kinetic energy per unit volume were therefore

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before the flow began, the whole energy of molecular motion and flow in the exit pipe, viz.

K = pG2 + pa2,

must be the same as before.

We have, therefore,

G2 = G2 — a2,

or the molecular speed G of the flowing gas is less than the molecular speed G, of the gas at rest. The cross

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of the gas when flowing is, therefore, less than the pressure when the gas was at rest. This lowering of the pressure by the flow depends, as the formulæ show, on cooling being produced.

Since these formulæ contain the velocity only in its square, they are independent of the direction of the motion, and hold, therefore, as well for to-and-fro oscillations as for the propagation of the longitudinal waves of sound. On this depend the apparent attractions and repulsions in air when sounding and in the ribbed dust-figures of Kundt.'

36. Propagation of Sound

When we develop the theory of sound according to the kinetic hypothesis we have also to consider two sorts of motion which exist without disturbing each other. In addition to the molecular motion which is present even in a gas at rest there are the to-and-fro motions which constitute

1 See W. König, Wied. Ann. xlii. 1891, p. 353; Zeitschr. f. phys. u. chem. Unterr. 8. Jahrg. 1895, p. 191.

the vibrations of sound. The latter motions spread from one place to another, and the cause of this transmission is the molecular motions which bring the particles that execute the sound-vibrations into contact with others. From this it follows that the velocity of propagation of sound cannot depend on the nature of the sound-vibrations, but only on the molecular motions.

1

If we paid no regard to the variations in temperature which a gas undergoes by condensation or rarefaction, it would be easy to answer the question as to the speed with which, on the basis of the assumptions of the kinetic theory, a sound wave is propagated. If sound consists in alternate rarefactions and condensations of the air, the speed of its propagation cannot be different from the speed with which any inequality of the pressure that arises at any place would spread through air-filled space. Now, according to our theory the pressure arises from the to-and-fro motions of the particles, and is exerted and carried on from one layer to another by the same cause; the velocity with which a pressure- cr sound-wave is propagated must therefore be just as great as that with which the particles of gas move to and fro in the direction of propagation of the wave. The value of the component of the molecular motion in the given direction, and not the resultant velocity of the particles, comes therefore into account in the calculation of the velocity of sound; and hence it follows at once that the speed of propagation of sound in a gas must be smaller than the mean speed of the molecular motion in this gas. This theoretically deduced proposition is completely confirmed by experiment; for instance, in atmospheric air at 0° C. the speed of sound is about 332 metres per second, and is consequently considerably less than the mean molecular speeds G 485 and 2 447 (§ 28).

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How much smaller the speed of sound is we may easily, and with sufficient exactness, determine in the same way as in 10 we calculated the pressure of a gas. If the energy

2

This conclusion has been experimentally confirmed by Calderoni, Wied. Beibl. iii. p. 155.

2 More mathematically strict calculations have been made by Stefan (Pogg.

of the motion of the molecules in a given direction is onethird part of the whole energy, we might look on the magnitude

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as the mean value of the component in a given direction, G being, as before, the mean molecular speed; for, as the energy varies as the square of the velocity, the velocity varies as the square root of the energy. We therefore find for the velocity v of sound the formula

v = G√ 3,

which shows that v is less than G; we also have

v = Ων
8'

from which it appears that Maxwell's mean value of the molecular speed is also greater than that of sound. If we put these formulæ into the form

v2 = {G2 = }πN2 = p/p

we obtain Newton's1 formula, with which the speed of sound can be calculated from the pressure p and the corresponding density p of the gas.

But, as Laplace2 first saw, this formula needs a correction. The oscillations constituting sound depend not so much on the actual pressure and density of the gas as on the changes which they simultaneously undergo in consequence of the alternate condensations and rarefactions. With more correctness, therefore, should we have expressed G in terms of the variations which p and p undergo instead of in terms of p and p themselves, and this could have been Ann. cxviii. 1863, p. 494), Roiti (Mem. dell' Accad. dei Lincei [3] i. 1876, pp. 39, 762; Nuovo Cimento [2] xvi. 1876; [3] i. 1877, p. 42), and Brusotti (Ann. Scient. del Ist. Tecnico di Pavia, 1874-5, p. 171); further by Hoorweg (Arch. Néerl. xi. 1876, p. 131; Pogg. Beiblätter, i. 1877, p. 209), Mees and H. A. Lorentz (Versl. en Med. K. Akad. Amst. xv. 1880), Schlemüller (Die Fortpflanzungsgeschw. in einem theor. Gase, bearb. suf Grund d. dyn. Gastheorie, Prag 1894). S. T. Preston has given an elementary explanation of the process in Phil. Mag. [5] iii. 1877, p. 441.

1 Principia, ii. § 8, prop. 49, probl. 11.

2 Ann. Chim. Phys. [2] iii. 1816, p. 238; xx. 1822, p. 266; Méc. Cél. v.

done equally easily. For by Boyle's law, if p + dp and p+ dp represent the values of p and p when increased by compression, the ratio

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is constant so long as the temperature remains unchanged. We might therefore have written

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for the formula giving the speed of sound; and herein the increment dp of pressure and the corresponding increment dp of density may be taken either as finite or as infinitely small magnitudes.

But the ratio of the pressure to the density remains constant only so long as the temperature of the gas remains unaltered. If, however, a gas is made to occupy a smaller volume, not only do the pressure and density increase, but also the temperature; and if the gas expands, not only do its pressure and density diminish, but its temperature falls too. This rise and fall of temperature, when the volume undergoes change, have both the effect of causing the pressure to alter in greater measure than the density, and therefore the ratio of dp to dp has in the actual case a greater value than Boyle's law gives it when the temperature is not taken into account. Thus the formula must be completed by a factor which is greater than 1, and, according to the equations of the theory of heat, which have been established by Laplace and others, this factor is the ratio of the specific heat C at constant pressure to the specific heat c at constant volume. Consequently we have

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That with this improved formula, also, the ratio of v to N is less than 1, I will show by taking atmospheric air as an example. Putting for this substance

C = 1.405 c,

as is deduced from the observations that are discussed in

detail in § 55, we obtain

v = 0.74 N,

and, on substituting for n its value 447, as given in § 28,

v = 332 metres per second.

The ratio 0.74 calculated for air holds good equally for the other so-called permanent gases, and is also approximately admissible for those that are condensable, so that in general we may assume the ratio of the speed of sound to the mean molecular speed to be about in round numbers.

According to the formula, the speed v of sound must decrease with falling temperature, just as the molecular speed . This theoretical conclusion is confirmed by experiment. According to experiments made by Greely 2 in Arctic regions, at temperatures between -8° and — 48° C., the speed decreases by 0-603 metres per second with every degree; its value, therefore, may be represented as a function of the temperature 9 by the formula

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One directly valuable result of the numerical calculation of the mean speed with which the molecules of different gases move is obtained from the fact that from these numbers we can at once infer the speed with which gases will issue through fine openings in a thin wall. We have, therefore, to consider the phenomenon designated effusion Benzenberg, Gilb. Ann. xlii. 1812, p. 1.

2 Meteorolog. Zeitschrift, 7. Jahrg. 1890, p. 6 (25. Jahrg. d. Zeitschr. d. öst. Ges. für Met.).

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