τ = 2/k, σ = √(μ − 1⁄4k2),
(31)
and the constants a, ε are arbitrary. This may be described as a simple harmonic oscillation whose amplitude diminishes asymptotically to zero according to the law e−t/τ. The constant τ is called the modulus of decay of the oscillations; if it is large compared with 2π/σ the effect of friction on the period is of the second order of small quantities and may in general be ignored. We have seen that a true simple-harmonic vibration may be regarded as the orthogonal projection of uniform circular motion; it was pointed out by P. G. Tait that a similar representation of the type (30) is obtained if we replace the circle by an equiangular spiral described, with a constant angular velocity about the pole, in the direction of diminishing radius vector. When k2 > 4μ, the solution of (29) is, in real form,
x = a1e−t/τ1 + a2e−t/τ2,
(32)
where
1/τ1, 1/τ2 = 1⁄2k ± √(1⁄4k2 − μ).
(33)
The body now passes once (at most) through its equilibrium position, and the vibration is therefore styled aperiodic.
To find the forced oscillation due to a periodic force we have