Page:Encyclopædia Britannica, Ninth Edition, v. 1.djvu/121

of time, which was noted as accurately as possible, the sound of the bell transmitted through the air. The result was a velocity for the iron of $10.5$ time that in air. Similar experiments on iron telegraph wire, made more recently near Paris by Wertheim and Brequet, have led to an almost identical number. Unfortunately, owing to the metal in those experiments not forming a continuous whole, and to other causes, the results obtained, which fall short of those otherwise found, cannot be accepted as correct.

Other means therefore, of an indirect character, to which we will refer hereafter, have been resorted to for determining the velocity of sound in solids. Thus Wertheim, from the pitch of the lowest notes produced by longitudinal friction of wires or rods, has been led to assign to that velocity values ranging, in different metals, from ${\text{16,822 feet}}$ for iron, to $4030$ for lead, at temperature $68^{\circ }{\text{ Fahr.}}$, and which agree most remarkably with those calculated by means of the formula ${\text{V}}={\sqrt {\tfrac {e}{\rho }}}$. He points out, however, that these values refer only to solids whose cross dimensions are small in comparison with their length, and that in order to obtain the velocity of sound in an unlimited solid mass, it is requisite to multiply the value as above found by ${\sqrt {\tfrac {3}{2}}}$ or ${\tfrac {5}{4}}$ nearly. For while, in a solid bar, the extensions and contractions due to any disturbance take place laterally as well as longitudinally; in an extended solid, they can only occur in the latter direction, thus increasing the value of $e$.

27.To complete the discussion of the velocity of the propagation of sound, we have still to consider the case of transversal vibrations, such as are executed by the points of a stretched wire or cord when drawn out of its position of rest by a blow, or by the friction of a violin-bow.

Fig. 4.

Let $ox$ (fig. 4) be the position of the string when undisturbed, $mnp$ when displaced. We will suppose the amount of displacement to be very small, so that we may regard the distance between any two given points of it as remaining the same, and also that the tension ${\text{P}}$ of the string is not changed in its amount, but only in its direction, which is that of the string.

Take any origin $o$ in $ox$, and $ab=bc=\delta x$ ( very small quantity), then the perpendiculars $am,bn,cp$, are the displacements of $abc$. Let $k,l$ be the middle points of $mn,np$; then $kl$ (which $=mn$ or $ab$ very nearly) may be regarded as a very small part of the string acted on by two forces each $={\text{P}}$, and acting at $n$ in the directions $np,nm$. These give a component parallel to $ac$, which on our supposition is negligible, and another ${\text{F}}$ along $nb$, such that

$${\text{F}}={\text{P}}(\sin {\theta }-\sin {\theta }\,^{\prime }={\text{P}}.({\tfrac {nq}{mn}}-{\tfrac {pr}{nq}})={\text{P}}.{\tfrac {nq-pr}{\delta x}}.$$

Now if $c=$ a length of string of weight equal to ${\text{P}}$, and the string to be supposed of uniformed thickness and density, the weight of $kl={\tfrac {\text{P}}{c}}.kl={\tfrac {\text{P}}{c}}.\delta x$, and the mass $m$ of $kl={\tfrac {P}{gc}}.\delta x$.

Hence the acceleration $f$ in direction $nb$ is—

$$f={\tfrac {\text{F}}{m}}=gc{\tfrac {nq-pr}{\delta x^{2}}}.$$

If we denote $ma$ by $y$, $oa$ by $x$, and the time by $t$, we shall readily see that this equation becomes ultimately,

$${\tfrac {d^{2}y}{dt^{2}}}=gc{\tfrac {d^{2}y}{dx^{2}}}$$

which is satisfied by putting

$$y=\phi (x+{\sqrt {gc.}}t)+\psi (x-{\sqrt {gc.}}t)$$

where $\phi$ and $\psi$ indicate any functions.

Now we know that if for a given value of $t$, $x$ be increased by the length $\lambda$ of the wave, the value of $y$ remains unchanged; hence,

$$\phi (x+{\sqrt {gc.}}t)+\&{\text{c.}}=\phi (x+\lambda +{\sqrt {gc.}}t)\&{\text{c.}}$$

But this condition is equally satisfied for a given value of $x$, by increasing ${\sqrt {gc.}}t.$ by $\lambda$, i.e., increasing $t$ by ${\tfrac {\lambda }{\sqrt {gc}}}$. This therefore must $={\text{T}}$ (the time of a complete vibration of any point of the string). But ${\text{V}}={\tfrac {\lambda }{\text{T}}}$. Hence,

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

is the expression for the velocity of sound when due to very small transversal vibrations of a thin wire or chord, which velocity is consequently the same as would be acquired by a body falling through a height equal to one half of a length of the chord such as to have a weight equal to the tension.

The above may also be put in the form:—

$${\text{V}}={\sqrt {\tfrac {g{\text{P}}}{w}}},$$

where ${\text{P}}$ is the tension, and $w$ the weight of the unit of length of the chord.

28.It appears then that while sound is propagated by longitudinal vibrations through a given substance with the same velocity under all circumstances, the rate of its transmission by transversal vibrations through the same substance depends on the tension and on the thickness. The former velocity bears to the latter the ratio of ${\sqrt {l}}:{\sqrt {c}}$, (where $l$ is the length of the substance, which would be lengthened one foot by the weight of one foot, if we take the foot as our unit) or of ${\sqrt {\tfrac {l}{c}}}:1$, that is, of the square root of the length which would be extended one foot by the weight of $c$ feet, or by the tension, to $1$. This, for ordinary tensions, results in the velocity for longitudinal vibrations being very much in excess of that for transversal vibrations.

29.It is a well known fact that, in all but very exceptional cases, the loudness of any sound is less as the distance increases between the source of sound and the ear. The law according to which this decay takes place is the same as obtains in other natural phenomena, viz., that in an unlimited and uniform medium the loudness or intensity of the sound proceeding from a very small sounding body (strictly speaking, a point) varies inversely as the square of the distance.

Fig. 5.

This follows from considering that the ear ${\text{AC}}$ receives only the conical portion ${\text{OAC}}$ of the whole volume of sound emanating from ${\text{O}}$, and that in order that an ear ${\text{BD}}$, placed at a greater distance from ${\text{O}}$, may admit the same quantity, its area must be to that of ${\text{AC}}:$ as ${\text{OB}}^{2}:{\text{OA}}^{2}$. But if ${\text{A}}^{\prime }={\text{AC}}$ be situated at same distance as ${\text{BD}}$, the amount of sound received by it and by ${\text{BD}}$ (and therefore by ${\text{AC}}$ will be as the area of ${\text{A}}^{1}$ or ${\text{AC}}$ to that of ${\text{BD}}$. Hence, the intensities of the sound as

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