as a function of $\omega $. Because of this increasingly rapid variation in $\omega $, the contribution of this term to $R^{(1)*}(t)=\int
gf_1^{(1)}d\omega $ decays in time (see Appendix I), and it does so at the same time-asymptotic rate as the contribution from the second more rapidly exponentially decaying contribution in Eq. (16). Thus the order parameter magnitude decays away, but the distribution function $f_1^{(1)}$ can still have a component (the first term in Eq. (16)) due to the pulse that has not decayed away. A similar conclusion applies for $\xi >0$ provided that $\xi $ is substantially less than the damping for the second term in Eq. (16). This is the source of the ‘memory’ referred to in Sec. I. It is also worth noting that the first term in Eq. (16) can be thought of as the manifestation of the continuous spectrum of the Kuramoto problem, discussed in detail in Ref.  [@strogatz92]. Thus the echo phenomenon that we derive subsequently can be regarded as an observable macroscopic consequence of the continuous spectrum, where by ‘macroscopic’ we mean that the effect can be seen through monitoring of the order parameter without the necessity of other more detailed knowledge of 