Page:Encyclopædia Britannica, Ninth Edition, v. 1.djvu/117

circle at $m$, and therefore as being perpendicular to ${\text{C}}m$. Hence $g$, acting parallel to $CA$, being resolved along $C{\text{m}}$ and $m{\text{A}}$, the former component is counteracted by the tension of the string, and there remains as the only effective acceleration, the tangential component along $m{\text{A}}$, which, by the triangle of forces, is equal to $g\cdot {\tfrac {{\text{A}}m}{{\text{C}}m}}$ or ${\tfrac {g}{l}}\cdot {\text{A}}m$, and is therefore proportional to ${\text{A}}m$.

On this supposition of indefinitely small vibrations, the pendulum is isochronous; that is, the time occupied in passing from one extreme position to the other is the same, for a given length $l$ of the pendulum, whatever the extent of vibration.

We conclude from this that, whatever may be the nature of the forces by which a particle is urged, if the resultant of those forces is directed towards a fixed point, and is proportional to the distance from that point, the particle will oscillate to and fro about that point in times which are independent of the amplitudes of the vibrations, provided these are very small.

5.The particle, whose vibratory motion we have been considering, is a solitary particle acted on by external forces. But, in acoustics, we have to do with the motion of particles forming a connected system or medium, in which the forces to be considered arise from the mutual actions of the particles. These forces are in equilibrium with each other when the particles occupy certain relative positions. But, if any new or disturbing force act for a short time on any one or more of the particles, so as to cause a mutual approach or a mutual recession, on the removal of the disturbing force, the disturbed particles will, if the body be elastic, forthwith move towards their respective positions of equilibrium. Hence arises a vibratory motion to and fro of each about a given point, analogous to that of a pendulum, the velocity at that point being always a maximum, alternately in opposite directions. Thus, for example, if to one extremity of a pipe containing air were applied a piston, of section equal to that of the pipe, by pushing in the piston slightly and then removing it, we should cause particles of air, forming a thin section at the extremity of the pipe, to vibrate in directions parallel to its axis.

In order that a medium may be capable of molecular vibrations, it must, as we have mentioned, possess elasticity, that is, a tendency always to return to its original condition when slightly disturbed out of it.

6.We now proceed to show how the disturbance whereby certain particles of an elastic medium are displaced from m of their equilibrium-positions, is successively transmitted to the remaining particles of the medium, so as to cause these also to vibrate to and fro.

Let us consider a line of such particles $y,x,a,b,$ & $c$.

$$y\;x\;a_{1}\;a_{2}\;b\;c\;d\;e\;f\;g\;h\;i\;j\;k\;l\;m\;n\;o\;p$$

equidistant from each other, as above; and suppose one of them, say $a$, to be displaced, by any means, to $a_{1}$. As we have seen, this particle will swing from $a_{1}$ to $a_{2}$ and back again, occupying a certain time ${\text{T}}$, to complete its double vibration. But it is obvious that, the distance between $a$ and the next particle $b$ to the right being diminished by the displacement of the former to $a_{1}$, a tendency is generated in $b$ to move towards $a_{1}$, the mutual forces being no longer in equilibrium, but having a resultant in the direction $ba_{1}$. The particle $b$ will therefore also suffer displacement, and be compelled to swing to and fro about the point $b$. For similar reasons the particles $c,d\ldots$ will all likewise be thrown into vibration. Thus it is, then, that the disturbance propagates itself in the direction under consideration. There is evidently also, in the case supposed, a transmission from $a$ to $x,y,$ &c., i.e., in the opposite direction.

Confining our attention to propagation in the direction $abc\ldots .$, we have next to remark that each particle in that line will be affected by the disturbance always later than the particle immediately preceding it, so as to be found in the same stage of vibration a certain interval of time after the preceding particle.

7.Two particles which are in the same stage of vibration, that is, are equally displaced from their equilibrium-positions, and are moving in the same direction and with equal velocities, are said to be in the same phase. Hence we may express the preceding statement more briefly thus: Two particles of a disturbed medium at different distances from the centre of disturbance, are in the same phase at different times, the one whose distance from that centre is the greater being later than the other.

8.Let us in the meantime assume that, the intervals $ab,be,cd\ldots .$ being equal, the intervals of time which elapse between the like phases of $b$ and $a$, of $c$ and $b\ldots .$ are also equal to each other, and let us consider what at any given instant are the appearances presented by the different particles in the row.

${\text{T}}$ being the time of a complete vibration of each particle, let ${\tfrac {\text{T}}{p}}$ be the interval of time requisite for any phase of $a$ to pass on to $b$. If then at a certain instant $a$ is displaced to its greatest extent to the right, $b$ will be somewhat short of, but moving towards, its corresponding position, $c$ still further short, and so on. Proceeding in this way, we shall come at length to a particle $p$, for which the distance $ap=p.ab$, which therefore lags in its vibrations behind a by a time $=p\times {\tfrac {\text{T}}{p}}={\text{T}}$, and is consequently precisely in the same phase as $a$. And between these two particles $a,p,$ we shall evidently have particles in all the possible phases of the vibratory motion. At $h$, which is at distance from $a={\tfrac {1}{2}}ap$, the difference of phase, compared with $a$, will be ${\tfrac {1}{2}}{\text{T}}$, that is, $h$ will, at the given instant, be dis placed to the greatest extent on the opposite side of its equilibrium-position from that in which $a$ is displaced; in other words, $h$ is in the exactly opposite phase to $a$.

9.In the case we have just been considering, the vibrations of the particles have been supposed to take place in a direction coincident with that in which the disturbance passes from one particle to another. The vibrations are then termed longitudinal.

But it need scarcely be observed that the vibrations may take place in any direction whatever, and may even be curvilinear. If they take place in directions at right angles to the line of progress of the disturbance, they are said to be transversal.

10.Now the reasoning employed in the preceding case will evidently admit of general application, and will, in particular, hold for transversal vibrations. Hence if we mark (as is done in fig. 2) the positions $a_{1}\;b_{1};c_{1}\ldots$, occupied by the various particles, when swinging transversely, at the instant at which $a$ has its maximum displacement above its equilibrium-position, and trace a continuous line running through the points so found, that line will by its ordinates indicate to the eye the state of motion at the given instant.

Fig. 2.

Thus $a$ and $p$ are in the same phase, as are also $b$ and $q,c$ and $r$, &c. $a$ and $h$ are in opposite phases, as are also $b$ and $i,c$ and $k$, &c.