Page:Encyclopædia Britannica, Ninth Edition, v. 1.djvu/120

If $\gamma$ were equal to $1,{\text{V}}={\text{U}}$, which is the result obtained by Newton, and would indicate that the velocity of sound in a gas equals the velocity of a body falling from a height equal to half of that of a homogeneous atmosphere of the gas.

20.In common dry air at $32^{\circ }{\text{ Fahr.}}$, $g$ being ${\text{32.2 ft.}}$, and the mercurial barometer ${\text{30 ins.}}$ or ${\text{2.5 ft.}}$, the density of air is to that of mercury as $1:{\text{10,485.6}}$; hence ${\text{H}}={\text{10,458.6}}\times {\text{2.5 ft.}}={\text{26,214 ft.}}$

Also $\gamma =1.408$ Hence $V_{0}={\sqrt {1,408\times 32,2\times 26,214}}={\text{1090 ft.}}$

and, by § 18, the increase of velocity for each degree of rise of temperature ($\alpha$ being ${\tfrac {1}{491}}$) is ${\tfrac {1099}{982}}$ or ${\tfrac {545}{491}}={\text{1.110 ft.}}$, $1{\tfrac {1}{9}}{\text{ ft.}}$ very nearly.

21.If the value of $\gamma$ were the same for different gases, it is obvious from formula ${\text{V}}={\sqrt {\gamma {\tfrac {p}{\rho }}}}$ that, at a given temperature, the velocities of sound in those gases would be to each other inversely as the square roots of their densities. Regnault has found that this is so for common air, carbonic acid, nitrous oxie, hydrogen and ammoniacal gas (though less so as regards the two last).

22.The experimental determination of the velocity of sound in air has been carried out by ascertaining accurately the time intervening between the flash and report of a gun as observed at a given distance, and dividing the distance by the time. A discussion of the many experiments conducted on this principle in various countries and at various periods, by Van Der Kolk (Lond. and Edin. Phil. Mag., July 1865), assigns to the velocity of sound in dry air at $32^{\circ }{\text{ Fahr.}}$, ${\text{1091 ft. 8 in.}}$ per second, with a probably error of $\pm {\text{3.7 ft.}}$; and still more recently (in 1871) Mr Stone, the Astronomer Royal at the Cape of Good Hope, has found 1090.6 as the result of careful experiments by himself there. The coincidence of these numbers with that we have already obtained theoretically sufficiently establishes the general accuracy of the theory.

23.Still it cannot be overlooked that the formula for ${\text{V}}$ is founded on assumptions which, though approximately, are not strictly correct. Thus, the air is not a perfect gas, nor is the variation of elastic force, caused by the passage comparison with the elastic force of the undisturbed air. Earnshaw (1858) first drew attention to these points, and came to the conclusion that the velocity of sound increases with its loudness, that is, with the violence of the disturbance. In confirmation of this statement, he appeals to a singular fact, viz., that, during experiments made by Captain Parry, in the North Polar Regions, for determining the velocity of sound, it was invariably found that the report of the discharge of cannon was heard, at a distance of $2{\tfrac {1}{2}}$ miles, perceptibly earlier than the sound of the word fire, which, of course, preceded the discharge.

As, in the course of propagation in unlimited air, there is a gradual decay in the intensity of sound, it would follow that the velocity must also gradually decrease as the sound proceeds onwards. This curious inference has been verified experimentally by Regnault, who found the velocity of sound to have decreased by ${\text{2.2 ft.}}$ per second in passing from a distance of $4000$ to one of ${\text{7500 feet}}$.

24.Among other interesting results, derived by the accurate methods adopted by Regnault, but which want of space forbids us to describe, may be mentioned the dependence of the velocity of sound on its pitch, lower notes being, cæt. par., transmitted at a more rapid rate than higher ones. Thus, the fundamental note of a trumpet travels faster than its harmonies.

25.The velocity of sound in liquids and solid (the displacements being longitudinal), may be obtained by formula (I.), neglecting the thermic effects of the compressions and expansions as being comparatively inconsiderable, and may be put in other forms:

Thus, if we denote by $\varepsilon$ the change in length of one foot of a column of the substance produced by its own weight $w$, then $e$ being $={\tfrac {w}{\varepsilon }}$ or ${\tfrac {g\rho }{\varepsilon }}$, we have ${\tfrac {e}{\rho }}={\tfrac {g}{\varepsilon }}$ and hence:

$${\text{V}}={\sqrt {\tfrac {g}{\varepsilon }}}$$

or, replacing ${\tfrac {1}{\varepsilon }}$ (which is the length in feet of a column that would be increased ${\text{1 foot}}$ by the weight of ${\text{1 cubic foot}}$) by $l$,

$${\text{V}}={\sqrt {gl}}$$

which shows that the velocity is that due to a fall through ${\tfrac {l}{2}}$.

Or, again, in the case of a liquid, if $\nu$ denote the change of volume, which would be produced by an increase of pressure equal to one atmosphere, or to that of a column ${\text{H}}$ of the liquid, since $\varepsilon$ is the change of volume due to weight of a column ${\text{H}}$ of the liquid, and $\therefore {\tfrac {\nu }{\varepsilon }}={\tfrac {\text{H}}{1}}$ and ${\tfrac {1}{\varepsilon }}={\tfrac {\text{H}}{\nu }}$, we get

$${\text{V}}={\sqrt {\tfrac {g{\text{H}}}{\nu }}}$$

Ex. 1.For water, ${\tfrac {1}{\nu }}={\text{20,000}}$ very nearly; ${\text{H}}={\text{34 ft.}}$ and hence ${\text{V}}={\text{4680 feet}}.$

This number coincides very closely with the value obtained, whether by direct experiment, as by Colladon and Sturm on the Lake of Geneva in 1826, who found $4708$, or by indirect means which assign to the velocity in the water of the River Seine at $59^{\circ }{\text{ Fahr.}}$ a velocity of ${\text{4714 ft.}}$ (Wertheim).

Ex. 2.For iron. Let the weight necessary to double the length of an iron bar be ${\text{4260}}$ millions of lbs. on the square foot. Then a length $l$ will be extended to $l+1$ by a force of ${\tfrac {\text{4260 millions lbs.}}{l}}$ on the sq. ft. This, therefore, by our definition of $l$, must be the weight of a cubic foot of the iron. Assuming the density of iron to be $7.8$, and ${\text{62.32 lbs.}}$ as the weight of a cubic foot of water, we get $7.8\times 62.32$ or ${\text{486 lbs.}}$ as the weight of an equal bulk of iron. Hence ${\tfrac {\text{4260 millions}}{l}}=486$ and $l={\tfrac {4260}{486}}{\text{ millions}}$, which gives ${\text{V}}={\sqrt {gl}}={\sqrt {{\tfrac {32.2\times 4260}{486}}{\text{ millions}}}}$ $={\sqrt {\tfrac {4260}{15}}}\times 1000=1000{\sqrt {284}}$ or ${\text{V}}={\text{17,000}}$ feet per second nearly.

As in the case of water and iron, so, in general, it may be stated that sound travels faster in liquids than in air, and still faster in solids, the ratio ${\tfrac {e}{\rho }}$ being least in gases and greatest in solids.

26.Biot, about 50 years ago, availed himself of the great difference in the velocity of the propagation of sound through metals and through air, to determine the ratio of the one velocity to the other. A bell placed near one extremity of a train of iron pipes forming a joint length of upwards of ${\text{3000 feet}}$, being struck at the same instant as the same extremity heard first the sound of the blow on the pipe, conveyed through the iron, and then, after an interval