divided by the difference in the $\tau $ values, provides a potential means of estimating $\xi $. Also, as indicated by Eq. (43), if one can lower the coupling $K$ sufficiently, then echoes provide a potential way of determining the oscillator frequency distribution function $g(\omega )$. In particular, for low $K$ the distribution $g(\omega )$ is directly given by the inverse Fourier transform of the echo profile. On the other hand, we have seen from the simulations in Sec. IV that finite $N$ leads to noise-like behavior that may compromise such attempts. We also note that the Kuramoto model is an idealization, and application to any given situation may require modifications of the model and theory to more closely correspond to the situation at hand. We, nevertheless, feel that consideration of echoes for diagnostics may be of potential use.

Furthermore, these phenomena are of theoretical interest from at least two points of view. First, as mentioned in Sec. IIIb, the memory required by the echo phenomenon can be thought of as leading to a macroscopically observable consequence of the continuous spectrum[@strogatz92] of the Kuramoto model. A second point of theoretical interest relates to the recent work in Ref.  [@ott]. In 