i. Elementary View of the UNDULATORY Theory of LIGHT. By Mr. FRESNEL. [◊] [Continued from the last Number.]
I SHALL not undertake to explain here in detail the reasons and the calculations which lead to the general formulas that I have employed to determine the position of the fringes and the intensity of the inflected rays: but I think it right to give at least a distinct idea of the principles on which this theory rests, and particularly of the principle of interference, which explains the mutual action of the rays of light on each other. The name of interference was given by Dr. YOUNG to the law which he discovered, and of which he has made so many ingenious applications.
This singular phenomenon, so difficult to be satisfactorily explained in the system of emanation, is on the contrary so natural a consequence of the theory of undulation, that it might have been predicted from a general consideration of the principles of that theory. Every body must have observed, in throwing stones into a pond, that, when two groups of waves cross each other on its surface, there are points at which the water remains immoveable, when the two systems are nearly of the same magnitude, while there are other places in which the force of the waves is augmented by their concurrence. The reason of this is easily understood. The undulatory motion of the surface of the water consists of vertical motions, which alternately raise and depress the particles of the fluid. Now, in consequence of the intersection of the waves, it happens, that at certain points of their meeting, one of the two waves has an ascending motion belonging to it, while the other tends at the same instant to depress the surface of the liquid: consequently, when the two opposite impulses are equal, it can neither be actuated by one nor the other, but must remain at rest. On the contrary, at the points in which the motions agree in their direction, and conspire with each other, the liquid, urged in the same direction [p114] by each of the forces, is raised or depressed with a velocity equal to the sum of the effects of the two separate impulses, or to the double of either of them taken singly, since they are now supposed to be equal. Between these points of perfect agreement and complete opposition, which exhibit, one the total absence of motion, the other the maximum of oscillation, there are an infinity of intermediate points, at which the alternate motion takes place with more or less of energy, accordingly as they approach more or less to the places of perfect agreement, or of complete opposition of the two systems of motion which are thus combined, or superinduced on each other.
The waves which are propagated in the interior of an elastic fluid, though very different in their nature from those of a liquid like water, produce mechanical effects by their interference, which are exactly of the same kind, since they consist in alternate oscillatory motions of the particles of the fluid. In fact, it is sufficient that these motions should be oscillatory, that is, that the particles should be carried by them alternately in opposite directions, in order that the effects of one series of waves may be destroyed by those of another series of equal intensity; for, provided that the difference of the route of the two groups of waves [derived from the same origin] be such, that for each point of the fluid the motions in one direction, belonging to the first series, correspond to the motions, belonging to the second, in the opposite direction, they must perfectly neutralise each other, if their intensity is equal: and the particles of the fluid must remain in repose. This result will always hold good, whatever may happen to be the direction of the oscillatory motion, with regard to that in which the undulations are propagated; provided that the direction of the oscillatory motion be the same in the two series to be combined. In the waves which are formed on the surface of a liquid, for example, the direction of the oscillation is [principally] vertical, while the waves are propagated horizontally, and consequently in a direction perpendicular to the former; in the undulations of sound, on the contrary, the oscillatory motion is parallel to the direction of the propagation of the sound, [or rather is [p115] identical with it]; and these undulations, as well as the waves of water, are subject to the laws of interference.
The undulations formed in the interior of a fluid have here been mentioned in a general manner: in order to form a distinct idea of this mode of propagation, it must be remarked, that when the fluid has the same density and the same elasticity in every direction, the agitation produced in any point must be propagated on all sides with the same velocity: for this velocity of propagation, which must not be confounded with the absolute velocity of the particles, depends only on the density and elasticity of the fluid. It follows thence that all the points, agitated at the same instant in a similar manner, must be found in a spherical surface, having for its centre the point which is the origin of the agitation: so that these undulations are spherical, while the waves, which are seen on the surface of a liquid, are simply circular.
We give the name of rays to the right lines drawn from the centre of agitation to the different points of this spherical surface; and these rays are the directions in which the motion is propagated. This is the meaning of the term sonorous rays in acustics, and of luminous rays or rays of light in the system which attributes the phenomena of light to the vibrations of a universal fluid, to which the name of ether has been given.
The nature of the different elementary motions, of which each wave is composed, depends on the nature of the different motions which constitute the primitive agitation. The simplest hypothesis that can be entertained concerning the formation of the luminous undulations, is, that the small oscillations of the particles of the bodies, which produce them, are analogous to those of a pendulum removed but little from its point of rest; for we must conceive the particles of bodies, not as immoveably fixed in the positions which they occupy, but as suspended by forces which form an equilibrium in all directions. Now, whatever the nature of such forces may be, as long as the displacement of the particles is but small in proportion to the extent of their sphere of action, the accelerating force which tends to restore them to their natural position, and which thus causes them to oscillate on each side of it, may always, without sensible error, be considered as proportional [p116] to the magnitude of that displacement: so that the law of their motion must be the same as that of the motion of the pendulum, and of all small oscillations in general. This hypothesis, which is suggested by the analogy with other natural phenomena, and which is the simplest that can be formed respecting the vibrations of the luminous particles, may be considered as experimentally confirmed by the observation, that the optical properties of light are all independent of any circumstances which cause the greatest difference in the intensity of the vibrations: so that the law of their motion must be presumed to be the same for the greatest as for the smallest.
It follows from this hypothesis respecting the small oscillations, that the velocity of the vibrating particle at each instant is proportional to the sine of an arc, representing the time elapsed from the beginning of the motion, taking the circumference for the whole time required for the return of the particle to the same point, that is, the time occupied by two oscillations, the one forwards and the other backwards. Such is the law according to which I have calculated the formulas which serve to determine the effect of any number of systems of waves of which the intensities and the relative positions are given. These formulas will be found in the Annals of Chemistry, vol. xi., page 254: [they may be applied with security to the phenomena there considered, though the perfect accuracy of the hypothesis in all possible cases may be questioned, upon the grounds of the microscopical observations on the motions of vibrating chords, published by Dr. Young in the Philosophical Transactions for 1800. TR.] Without entering into the details of the calculations, I think it necessary to show in what manner the nature of the undulation depends on the kind of motion of the vibrating particles.
Let us suppose, in the fluid, a little solid plane which is removed from its primitive position, towards which it is urged by a force proportional to the distance. At the beginning of its motion, the accelerative force produces in it an infinitely small velocity only; but its action continuing, the effects become accumulated, and the velocity of the solid plane goes on continually to increase, until the moment of its arrival at [p117] the position of equilibrium, in which it would remain, but for the velocity which it has acquired; and it is by this velocity only, that it is carried beyond the point of equilibrium. The same force which tends towards this point, and which now begins to act in a contrary direction, continually diminishes the velocity, until it is completely annihilated; and then the force continuing its action produces a velocity in the contrary direction, which brings the plane back to its place of equilibrium. This velocity again is very small at the commencement of the return of the particle, or plane, and increases by the same degrees as it had before diminished, until the instant of the arrival of the particle at the neutral point, which it passes with the velocity previously acquired: but when it has passed this point, the motion is diminished more and more by the effect of the force tending towards it, and its velocity is reduced to nothing when it arrives at the place of the commencement of the motion. It then recommences, at similar periods, the series of motions which have been described, and would continue to oscillate for ever, but for the effect of the resistance of the surrounding fluid, the inertia of which continually diminishes the amplitude of its oscillations, and finally extinguishes them at the end of a longer or shorter time, according to circumstances. [It must not be inferred from this explanation, that the particles of a fluid transmitting an undulation have any tendency to vibrate for ever: on the contrary it has been admitted by the best writers on the theory of sound, that all the motions which constitute it, as considered in a fluid, are completely transitory in their nature, and have no disposition to be repeated after having been once transmitted to a remoter part of the fluid. TR.]
Let us now consider in what manner the fluid is agitated by these oscillations of the solid plane. The stratum immediately in contact with it, being urged by the plane, receives from it at each instant the velocity of its motion, and communicates it to the neighbouring stratum, which it forces forwards in its turn, and from which the motion is communicated successively to the other strata of the fluid; but this transmission of the motion is not instantaneous, and it is only at the end of a certain time that it arrives at a determinate [p118] distance from the centre of agitation. This time is the shorter, as the fluid is less dense, and more elastic; that is, composed of particles which possess a greater repulsive force. This being granted, let us assume, in order to facilitate the explanation, the moment when the moveable plane is returned to the initial situation, after having performed two complete oscillations in opposite directions: at this moment, the nascent velocity, which it had at first, is transmitted to a stratum of the fluid removed from the centre of agitation by a distance which we may represent by d. Immediately afterwards, the velocity of the moveable plane, which has a little augmented, has been communicated to the stratum in contact with it: “hence it has passed successively through all the following strata;” and at the moment when the first agitation arrives at the stratum of which the distance is d, the second has arrived at the stratum immediately before it. Continuing thus to divide, in our imagination, the duration of the two oscillations of the moveable plane into an infinity of small intervals of time, and the fluid comprehended in the length d, into an equal number of infinitely thin strata, it is easy to perceive, by the same reasoning, that the different velocities of the moveable plane, at each of these instants, are now distributed among the corresponding strata; and that thus, for example, the velocity which the plane possessed at the middle of the first oscillations in the direction of the motion, must have arrived, at the instant in question, at the distance 34 d: so that it is the stratum at this distance which possesses at the moment the greatest direct velocity; and in the same manner when the plane arrived at the limit of its first direct oscillation, its velocity was extinguished, and the same absence of motion will be found at the distance 12 d.