(10)
where n = √μ. Unless the initial conditions be adjusted so as to make A = 0 exactly, x will ultimately increase indefinitely with t. The position x = 0 is one of equilibrium, but it is unstable. This applies to the inverted pendulum, with μ = g/l, but the equation (9) is then only approximate, and the solution therefore only serves to represent the initial stages of a motion in the neighbourhood of the position of unstable equilibrium.
In acoustics we meet with the case where a body is urged towards a fixed point by a force varying as the distance, and is also acted upon by an “extraneous” or “disturbing” force which is a given function of the time. The most important case is where this function is simple-harmonic, so that the equation (5) is replaced by
| d2x | + μx = ƒ cos (σ1t + α), |
| dt2 |
(11)
where σ1 is prescribed. A particular solution is
| x = | ƒ | cos (σ1t + α). |
| μ − σ12 |
(12)
This represents a forced oscillation whose period 2π/σ1, coincides with that of the disturbing force; and the phase agrees with that of the force, or is opposed to it, according as σ12 < or > μ; i.e. according as the imposed period is greater or less than the natural period 2π/√μ. The solution fails when the two periods agree exactly; the formula (12) is then replaced by
| x = | ƒt | sin (σ1t + α), |
| 2σ1 |