The Quarterly Journal of Science, Literature and the Arts, July-December, 1827

by Various

It is always supposed, that the oscillations of the plane are so minute in comparison with the length d, that their extent may be neglected in this calculation: and this hypothesis is actually consistent with the fact, since there is every reason to suppose that the excursions of the incandescent particles are very small in comparison with the extent of an undulation, which, though an extremely minute space, is still an appreciable quantity, and may be actually measured. Besides, [p119] even if the amplitude of these oscillations were not in the first instance so wholly inconsiderable, it would be sufficient to consider an undulation at a greater distance from the centre of agitation, in order that their extent might be diminished in any required proportion.

In the second, or retrograde oscillation, the plane, returning through the same space, must communicate to the stratum of fluid in contact with it, and to the rest in succession, a motion in a direction contrary to that of the first oscillation; for when the plane recedes, the stratum in contact with it, urged against the plane by the elasticity or the expansive force of the fluid, necessarily follows it, and fills up the vacuum which its retrograde motion tends to produce. For the same reason, the second stratum is urged against the first, the third against the second, and so forth. It is thus that the retrograde motion is communicated, step by step, to the most distant strata: its propagation is effected according to the same law that governs the direct motion; the only difference is in the direction of the motions, or, in the language of mathematics, in the sign of the velocities which are imparted to the molecules of the fluid. We see then that the different velocities which have existed in the solid plane, during its second oscillation, must exist at the moment which we are considering, in the different strata comprehended in the other half of d, but with contrary signs. Thus the velocity, for example, which the plane had in the middle of the second oscillation, which is its maximum of retrograde velocity, must now be found in the fluid stratum situated at the distance 14 d from the centre of agitation, while the maximum of direct velocity is found, at the same instant, in the stratum which is at the distance 34 d from the centre of agitation.

The extent of the fluid, agitated by the two opposite oscillations of the solid plane, is what we call the breadth of an entire undulation, and we may consequently give the name of semiundulation to each of the parts actuated by the opposite undulations; the whole constituting a complete oscillation, since it comprehends the return of the vibrating plane to the initial situation. It is obvious, that the two semiundulations, which compose the complete undulation, exhibit, in [p120] the fluid strata which they contain, velocities absolutely equal in magnitude, but with contrary signs, that is to say, carrying the particles of the fluid in opposite directions. These velocities are the greatest in the middle of each of the semiundulations, and decrease gradually towards their extremities, where they entirely vanish: so that the points of rest, and of the greatest velocities positive and negative, are separated from each other by intervals of one fourth of an undulation.

The length of an undulation, d, depends on two things: first, on the promptitude with which the motion is propagated in the fluid; and secondly, the duration of the complete oscillation of the vibrating plane; for the longer this duration, and the more rapid the propagation of the motion, the greater will be the distance to which the first agitation has been extended at the instant of the return of the solid plane to its initial situation. If the oscillations are all performed in the same medium, the velocity of propagation remaining the same, the length of the undulations will be simply proportional to the duration of the oscillations of the vibrating particles from which they originate. As long as the vibrating particles continue to be subjected to the same forces, it follows from the principles of mechanics that each of their minute oscillations will occupy the same time, whatever their extent may be; so that the corresponding undulations of the fluid will continue to be of the same length; they will only differ from each other in the greater or less extent of the elementary vibrations of the particles, which will be proportional to the extent of the luminous particles; for it appears from what has already been stated, that each stratum of the fluid repeats exactly all the motions of the vibrating particle. The greater or less amplitude of the oscillations of the strata of the fluid determines the degree of absolute velocity with which they move, and consequently the energy, but not the nature of the sensation which they excite, which must depend, according to every analogy, upon the duration of the oscillations. It is thus that the nature of the sounds, transmitted by the air to our ears, depends entirely on the duration of each of the oscillations executed by the air, or by the sonorous [p121] body which puts it in motion; and that the greater or less amplitude or energy of the oscillations only augments or diminishes the intensity of the sound, without changing its nature, that is, its tone, or pitch.

The intensity of the light must depend then on the intensity of the vibrations of the ether; and its nature, that is to say, the sensation of colour that it produces, will depend on the duration of each oscillation, or on the length of the undulation, the one of these being proportional to the other. [We find, however, nothing in light of the same colour that is at all analogous to the different register, quality, or timbre of a sound; by which, for instance, the sound of a violin differs from that of a flute in unison with it: the subordinate, or harmonic tones of the sound having nothing in light to correspond with them. TR.]

The duration of the elementary oscillation remaining the same, the absolute velocity of the ethereal particles, at the corresponding periods of the oscillatory motions, is, as we have seen, proportional to its extent. It is the square of this velocity, multiplied by the density of the fluid, that represents what is called the living force in mechanics, or otherwise the energy or impetus of the particles, which is to be taken as the measure of the sensation produced, or of the intensity of the light: thus, for example, if in the same medium, the amplitude of the oscillation is doubled, the absolute velocities will also be doubled, and the living force, or the intensity of the light, will be quadrupled.

We must, however, take care not to confound this absolute velocity of the particles of the fluid with the velocity of the propagation of the agitation. The first varies according to the amplitude of the oscillations; the second, which is nothing but the promptitude with which the motion is communicated from one stratum to the other, is independent of the intensity of the vibrations. It is for this reason, that a weak sound is transmitted by the air with the same velocity as a stronger one; and that the least intense light is propagated with the same rapidity as the brightest. When we speak of the velocity of light, we always speak of the velocity of its propagation. Thus, when we say that light passes through 200 thousand [p122] miles in a second, we do not mean, according to the undulatory system, that such is the absolute velocity of the ethereal particles; but that the motion communicated to the ether employs only a second to pass to a stratum at the distance of 200 thousand miles from its origin.

In proportion as the undulation becomes more distant from the centre of agitation, the motion, spreading over a greater distance, must be weakened in every part of the wave. It is shown by calculation, that the amplitude of the oscillatory motion, or the absolute velocity of the particles concerned in it, is inversely proportional to the distance from the centre of agitation. Consequently, the square of this velocity is inversely proportional to the square of the distance, and the intensity of the light must be inversely as the square of the distance from the luminous point. It must be remarked, that, for the same reasons, the sum of the living forces of the whole undulation remains unaltered; for, on one side the length of the undulation d, which may also be called its thickness, is invariable, and its extent of surface augmenting in proportion to the square of the distance from the centre, the quantity, or mass of the fluid agitated, is proportional to the same square: and since the squares of the absolute velocities are diminished in the same proportion as the masses have augmented, it follows that the sum of the products of the masses by the squares of the velocities, that is to say, the sum of the living forces, remains unaltered. It is a general principle of the motion of elastic fluids, that however the motion may be extended or subdivided, the total sum of the living forces remains constant; and this is the principal reason why the living force must be considered as the measure of light, of which the total quantity always remains very nearly the same, at least as long as it continues to pass through perfectly transparent mediums.

It may be remarked, that black substances, and even the most brilliant metallic surfaces, by no means reflect the whole of the light which falls on them; bodies which are imperfectly transparent, and even the most transparent, when of great thickness, absorb also, to use a common expression, a considerable portion of the light that is passing through [p123] them: but it must not be inferred that the principle of living forces is inapplicable to these phenomena; it follows, on the contrary, from the most probable idea that can be formed of the mechanical constitution of bodies, that the sum of the living force must remain always the same, as long as the accelerating forces tending to bring the particles to their natural positions remain unchanged, and that the quantity of living force which disappears in the state of light, instead of being annihilated, is reproduced in the form of heat.

In order to obtain a correct idea of the manner in which the oscillation of a small solid body occasions undulations in an elastic fluid, it has been only necessary to consider a complete oscillation of the solid plane, which produces an entire undulation. If we suppose the oscillations of the plane to be continually repeated, we shall have a series of undulations instead of a single one: and they will follow each other without intermission, provided that the vibrations of the particle first agitated have been regular. Such a series of regular and uninterrupted luminous motions I call a system of undulations.