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.]