$K=1=\frac{1}{2}K_c$, $h(\theta )=\sin \theta
+\sin 2\theta +\sin 3\theta $. In this case, echoes are seen at $t=2\tau =200$ and at $t=3\tau =300$. The inset shows a blow-up of the numerical result for the echo shape at $t=3\tau $ with the theoretical result, Eq. (41), superposed (dotted curve).](fig5)

Finally, we note that similar numerical experiments to all of the above have been repeated using a Gaussian $g(\omega )$, and these yield similar results (not shown).

V. Discussion
=============

Echo phenomena as used for MRI provide a powerful medical diagnostic tool. Echoes in plasmas have also been used as a basis for measuring velocity space diffusion of plasma particles[@jensen69]. Thus it is of interest to consider whether there are potential diagnostic measurement uses of echoes in the context of situations that can be described by the Kuramoto model and its variants. For example, we note that the amplitude of the echo varies exponentially with $\xi $, providing a possible means of determining the phase diffusion coefficient $\xi $. For example, the amplitude of the echo at $t=2\tau $ varies as $e^{-\xi \tau }$. Thus the log of the ratio of measurements of the echo amplitude using two different values of $\tau $, 