A note on the method of multiple scales
A modification of the method of multiple scales makes use of the expansion of parameters of the system in order to remove undesirable characteristics of the solution obtained by the usual multiple-scale method. When applied to the damped harmonic oscillator, the modification leads to the exact solution. For the Duffing equation it leads to an approximation which can be reduced to the solution reported by Nayfeh [1], However, the solution derived here appears to be more accurate and the frequency takes on a form without nonuniformities. The method of multiple scales produces a uniformly valid expansion for systems in which a troublesome term is multiplied by a small parameter and for which an ordinary perturbation expansion leads to a nonuniformly valid series solution. For example, the weakly damped harmonic oscillator 5c + 2ex -Ia>2x = 0, e <| 1, (1) has the general solution x = [a0ei^T~^i' + (2) where a* is the complex conjugate of a0. An ordinary perturbation in e yields results which correspond to expanding e~a and exp[ + i(co2 — e2)1/2f in power series which are nonuniformly valid since t can always be large enough to offset the smallness of e or s2. The derivative expansion method of multiple scales [1, pp. 236-240] makes use of the time expansion tn = e"', x = x(r0, tu t2 •• ), dx/dt = D0x + sDtx + e2D2x + ■■■, (3) where Dn = d/dtn. The time t0 is the unstretched time coordinate and the remaining tn correspond to longer time scales. To 0(s2) the solution