Page:Encyclopædia Britannica, Ninth Edition, v. 1.djvu/118

Distances $a\;p,b\;q,$ &c., separating particles in the same phase, and each of which, as we have seen, is passed over by the disturbance in the time ${\text{T}}$ of a complete vibration, include within them all the possible phases of the motion.

Beyond this distance, the curve repeats itself exactly, that is, the phases recur in the same order as before.

Now the figure so traced offers an obvious resemblance to the undulating surface of a lake or other body of water, after it has been disturbed by wind, exhibiting a wave with its trough ${\text{A}}h_{1}{\text{B}}$, and its crest ${\text{B}}p_{1}{\text{C}}$. Hence have been introduced into Acoustics, as also into Optics, the terms wave and undulation. The distance $ap$, or $bq\ldots$ or ${\text{A C}}$, which separates two particles in same phase, or which includes both a wave-crest and a wave-trough, is termed the length of the wave, and is usually denoted $\lambda$.

As the curve repeats itself at intervals each $=\lambda$, it follows that particles are in the same phase at any given moment, when the distances between them in the direction of transmission of the disturbance $=\lambda ,2\lambda ,3\lambda \ldots$ and generally $=n\lambda$, where $n$ is any whole number.

Particles such as $a$ and $h$, $b$ and $i$, &c., which are at distances ${\tfrac {1}{2}}\lambda$, being in opposite phases, so will also be particles separated by distance, ${\tfrac {1}{2}}\lambda +\lambda ={\tfrac {3}{2}}\lambda$,or, in general, by ${\tfrac {1}{2}}\lambda +m\lambda =(2m+1){\tfrac {\lambda }{2}}$, that is, by any odd multiple of ${\tfrac {\lambda }{2}}$

11.A like construction to the one just adopted for the displacements of the particles at any given instant, may be also applied for exhibiting graphically their velocities at the same instant. Erect at the various points $a,b,c,$ &c., perpendiculars to the line joining them, of lengths proportional to and in the direction of their velocities, and draw a line through the extreme points of these perpendiculars; this line will answer the purpose required. It is indicated by dots in the previous figure, and manifestly forms a wave of the same length as the wave of displacements, but the highest and lowest points of the one wave correspond to the points in which the other wave crosses the line of equilibrium.

12.In order to a graphic representation of the displacements and velocities of particles vibrating longitudinally, it is convenient to draw the lines which represent those quantities, not in the actual direction in which the motion takes place and which coincides with the line $a\;b\;c\ldots$, but at right angles to it, ordinates drawn upwards indicating displacements or velocities to the right (i.e., in the direction of transmission of the disturbance), and ordinates drawn downwards indicating displacements or velocities in the opposite direction. When this is done, waves of dis placement and velocity are figured identically with those for transversal vibrations, and are therefore subject to the same resulting laws.

13.But not only will the above waves enable us to see at a glance the circumstances of the vibratory motion at the instant of time for which it has been constructed, but also for any subsequent moment. Thus, if we desire to consider what is going on after an interval ${\tfrac {\text{T}}{p}}$, we have simply to conceive the whole wave (whether of displacement or velocity) to be moved to the right through a distance $=a\;b$. Then the state of motion in which a was before will have been transferred to $b$, that of $b$ will have been transferred to $c$, and so on. At the end of another such interval, the state of the particles will in like manner be represented by the wave, if pushed onward through another equal space. In short, the whole circumstances may be pictured to the eye by two waves (of displacement and of velocity) advancing continuously in the line $a\;b\;c\ldots$ with a velocity ${\text{V}}$ which will take it over the distance $ab$ in the time ${\tfrac {\text{T}}{p}},{\text{V}}$ being therefore $={\tfrac {\tfrac {ab}{\text{T}}}{p}}={\tfrac {p\cdot ab}{\text{T}}}={\tfrac {\text{AC}}{\text{T}}}$ or ${\text{V}}={\tfrac {\lambda }{\text{T}}}$.

This is termed the velocity of propagation of the wave, and, as we see, is equal to the length of the wave divided by the time of a complete vibration of each particle.

If, as is usually more convenient, we express ${\text{T}}$ in terms of the number $n$ of complete vibrations performed in a given time, say in the unit of time, we shall have ${\tfrac {1}{\text{T}}}=n$, and hence

$${\text{V}}=n\lambda$$

14.There is one very important distinction between the two cases of longitudinal and of transversal vibrations which now claims our attention, viz., that whereas vibrations of the latter kind, when propagated from particle to particle in an elastic medium, do not alter the relative distances of the particles, or, in other words, cause no change of density throughout the medium; longitudinal vibrations, on the other hand, by bringing the particles nearer to or further from one another than they are when undisturbed, are necessarily accompanied by alternate condensations and rarefactions.

Thus, in fig. 2, we see that at the instant to which that fig. refers, the displacements of the particles immediately adjoining $a$ are equal and in the same direction; hence at that moment the density of the medium at a is equal to that of the undisturbed medium. The same applies to the points $h,p,$ &c., in which the displacements are at their maxima and the velocities of vibration $=0$.

At any point, such as $c$, between $a$ and ${\text{A}}$, the displacements of the two adjoining particles on either side are both to the right, but that of the preceding particle is now the greater of the two, and hence the density of the medium throughout $a{\text{A}}$ exceeds the undisturbed density. So at any point, such as $f$, between ${\text{A}}$ and $h$, the same result holds good, because now the displacements are to the left, but are in excess on the right side of the point $f$. From $a$ to $h$, therefore, the medium is condensed.

From $h$ to ${\text{B}}$, as at $k$, the displacements of the two particles on either side are both to the left, that of the preceding particle being, however, the greater. The medium, therefore, is here in a state of rarefaction. And in like manner it may be shown that there is rarefaction from ${\text{B}}$ to $p$; so that the medium is rarefied from $h$ to $p$.

At ${\text{A}}$ the condensation is a maximum, because the displacements on the two sides of that point are equal and both directed towards ${\text{A}}$. At ${\text{B}}$, on the other hand, it is the rarefaction which is a maximum, the displacements on the right and left of that point being again equal, but directed outwards from ${\text{B}}$.

It clearly follows from all this that, if we trace a curve of which any ordinate shall be proportional to the difference between the density of the corresponding point of the disturbed medium and the density of the undisturbed medium—ordinates drawn upwards indicating condensation, and ordinates drawn downwards rarefaction—that curve will cross the line of rest of the particles $a\;b\;c\ldots$ in the same points as does the curve of velocities, and will therefore be of the same length $\lambda$, and will also rise above that line and dip below it at the same parts. But the connection between the wave of condensation and rarefaction and the wave of velocity, is still more intimate, when the extent to which the particles are displaced is very small, as is always the case in acoustics. For it may be shown that then the degree of condensation or rarefaction at any point of the medium is proportional to the velocity of vibration at that point. The same ordinates, therefore, will repre-