Page:Encyclopædia Britannica, Ninth Edition, v. 1.djvu/119

sent the degrees of condensation, which represent the velocities, or, in other words, the wave of condensation and rarefaction may be regarded as coincident with the velocity wave.

Part II.
Velocity of propagation of waves of longitudinal disturbance through any elastic medium.

15.Sir Isaac Newton was the first who attempted to determine, on theoretical grounds, the velocity of sound in air and other fluids. The formula obtained by him gives, however, a numerical value, as regards air, falling far short of the result derived from actual experiment; and it was not till long afterwards, when Laplace took up the question, that complete coincidence was arrived at between theory and observation. We are indebted to the late Professor Rankine, of Glasgow (Phil. Trans. 1870, p. 277)^[1], for a very simple and elegant investigation of the question, which we will here reproduce in an abridged form.

Let us conceive the longitudinal disturbance to be propagated through a medium contained in a straight tube having a transverse section equal to unity, but of indefinite length.

Fig. 3. Let two transverse planes ${\text{A}}_{1}\;{\text{A}}_{2}$ (fig. 3) be conceived as moving along the interior of the tube in the same direction and with the same velocity ${\text{V}}$ as the disturbance-wave itself.

Let $u_{1}\;u_{2}$ be the velocities of displacement of the particles of the medium at ${\text{A}}_{1}\;{\text{A}}_{2}$ respectively, at any given instant, estimated in the same direction as ${\text{V}}$; and $\rho _{1}\;\rho _{2}$ the corresponding densities of the medium.

The disturbances under consideration, being such as preserve a permanent type throughout their propagation, it follows that the quantity of matter between ${\text{A}}_{1}$, and ${\text{A}}_{2}$ remains constant during the motion of these planes, or that as much must pass into the intervening space through one of them as issues from it through the other. Now at ${\text{A}}_{1}$ the velocity of the particles relatively to ${\text{A}}_{1}$ itself is ${\text{V}}-u_{1}$ inwards, and consequently there flows into the space ${\text{A}}_{1}\;{\text{A}}_{2}$ through ${\text{A}}_{1}$ a mass $({\text{V}}-u_{1})\rho _{1}$ in the unit of time.

Forming a similar expression as regards ${\text{A}}_{2}$, putting $m$ for the invariable mass through which the disturbance is propagated in the unit of time, and considering that if $\rho$ denote the density of the undisturbed medium, $m$ is evidently equal to ${\text{V}}\rho$, we have—

$$({\text{V}}-u_{1})\rho _{1}=({\text{V}}-u_{2})\rho _{2}={\text{V}}\rho =m$$

Now, $p_{1}\;p_{2}$ being the pressures at ${\text{A}}_{1},{\text{A}}_{2}$ respectively, and therefore $p_{2}-p_{1}$ the force generating the acceleration $u_{2}-u_{1}$, in unit of time, on the mass $m$ of the medium, by the second law of motion,

$$p_{2}-p_{1}=m(u_{2}-u_{1})$$

Eliminating $u_{1},u_{2}$ from these equations, and putting for, ${\tfrac {1}{\rho _{1}}},{\tfrac {1}{\rho _{2}}},{\tfrac {1}{\rho }}$ the symbols $s_{1},s_{2},s$ (which therefore denote the volumes of the unit of mass of the disturbed medium at ${\text{A}}_{1},{\text{A}}_{2}$, and of the undisturbed medium), we get:

$$m^{2}={\tfrac {p_{2}-p_{1}}{s_{1}-s_{2}}}{\text{ and V}}^{2}=s^{2}{\tfrac {p_{2}-p_{1}}{s_{1}-s_{2}}}$$

Now, if (as is generally the case in sound) the changes of pressure and volume occurring during the disturbance of the medium are very small, we may assume that these changes are proportional one to the other. Hence, denoting the ratio which any increase of pressure bears to the diminution of the unit of volume of the substance, and which is termed the elasticity of the substance, by $e$, we shall obtain for the velocity of a wave of longitudinal displacements, supposed small, the equation:

	${\text{V}}=$	${\sqrt {es}}$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,}$
or	${\text{V}}=$	${\sqrt {\frac {e}{\rho }}}$

16.In applying this formula to the determination of the velocity of sound in any particular medium, it is requisite, as was shown by Laplace, to take into account the thermic effects produced by the condensations and rarefactions which, as we have seen, take place in the substance. The heat generated during the sudden compression, not being conveyed away, raises the value of the elasticity above that which otherwise it would have, and which was assigned to it by Sir Isaac Newton.

Thus, in a perfect gas, it is demonstrable by the principles of Thermodynamics, that the elasticity $e$, which, in the undisturbed state of the medium, would be simply equal to the pressure $p$, is to be made equal to $\gamma p$, where $\gamma$ is a number exceeding unity and represents the ratio of the specific heat of the gas under constant pressure to its specific heat at constant volume.

Hence, as air and most other gases may be practically regarded as perfect gases, we have for them:

$${\text{V}}={\sqrt {\gamma ps}}={\sqrt {\tfrac {\gamma p}{\rho }}}$$

17.From this the following inference may be drawn:—The velocity of sound in a given gas is unaffected by change of pressure if unattended by change of temperature. For, by Boyle's law, the ratio ${\tfrac {p}{\rho }}$ is constant at a given temperature. The accuracy of this inference has been confirmed by recent experiments of Regnault.

18.To ascertain the influence of change of temperature on the velocity of sound in a gas, we remark that, by Gay Lussac's law, the pressure of a gas at different temperatures varies proportionally both to its density $\rho$ and to $1+\alpha t$, where $t$ is the number of degrees of temperature above freezing point of water (32° Fahr.), and $\alpha$ is the expansion of unit of volume of the gas for every degree above 32°.

If, therefore, $p,p_{0},\rho ,\rho _{0}$ denote the pressures and densities corresponding to temperatures $32^{\circ }+t^{\circ }$and $32^{\circ }$, we have:

$${\tfrac {p}{p_{0}}}={\tfrac {\rho }{\rho _{0}}}(1+\alpha t)$$

and hence, denoting the corresponding velocities of sound by ${\text{V}},{\text{V}}_{0}$, we get:

$${\tfrac {\text{V}}{{\text{V}}_{0}}}={\sqrt {(1+\alpha t)}}$$

whence, $\alpha$ being always a very small fraction, is obtained very nearly:

$${\tfrac {\text{V}}{{\text{V}}_{0}}}=1+{\tfrac {\alpha }{2}}t{\text{ and V}}-{\text{V}}_{0}={\tfrac {\alpha }{2}}t{\text{V}}_{0}$$

The velocity increases, therefore, by ${\tfrac {\alpha }{2}}{\text{V}}_{0}$ for every degree of rise of temperature above 32°.

19.The general expression for ${\text{V}}$ given in (II.) may be put in a different form: if we introduce a height ${\text{H}}$ of the gas, regarded as having the same density $\rho$ throughout and exerting the pressure $p$, then $p=g\rho {\text{H}}$, where $g$ is the acceleration of gravity, and there results:

$${\text{V}}={\sqrt {\gamma g{\text{H}}}}$$

Now ${\sqrt {g{\text{H}}}}$ or ${\sqrt {2g{\tfrac {\text{H}}{2}}}}$ is the velocity ${\text{U}}$ which would be acquired by a body falling in vacuo from a height ${\tfrac {\text{H}}{2}}$. Hence ${\text{V}}={\text{U}}\;{\sqrt {\gamma }}$.

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1. See also Maxwell, Theory of Heat, p. 203.