(13)
which represents a vibration of continually increasing amplitude. Since the equation (12) is in practice generally only an approximation (as in the case of the pendulum), this solution can only be accepted as a representation of the initial stages of the forced oscillation. To obtain the complete solution of (11) we must of course superpose the free vibration (6) with its arbitrary constants in order to obtain a complete representation of the most general motion consequent on arbitrary initial conditions.
A simple mechanical illustration is afforded by the pendulum. If the point of suspension have an imposed simple vibration ξ = a cos σt in a horizontal line, the equation of small motion of the bob is
| mẍ = −mg | x − ξ | , |
| l |
or
| ẍ + | gx | = g | ξ | . |
| l | l |
(14)
| Fig. 63. |
This is the same as if the point of suspension were fixed, and a horizontal disturbing force mgξ/l were to act on the bob. The difference of phase of the forced vibration in the two cases is illustrated and explained in the annexed fig. 63, where the pendulum virtually oscillates about C as a fixed point of suspension. This illustration was given by T. Young in connexion with the kinetic theory of the tides, where the same point arises.
We may notice also the case of an attractive force varying inversely as the square of the distance from the origin. If μ be the acceleration at unit distance, we have