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We consider a simple dynamical and relativistic model to explain the spectro-temporal structure often displayed by repeating fast radio bursts (FRBs). We show how this model can account for the downward frequency drift in a sequence of sub-bursts of increasing arrival time (the "sad trombone" effect) and their tendency for exhibiting a reduced pulse width with increasing frequency of observation. Most importantly, this model also predicts a systematic inverse relationship between the (steeper) slope of the frequency drift observed within a single sub-burst and its temporal duration. Using already published data for FRB 121102 we find and verify the relationship predicted by this model. We therefore argue that the overall behaviour observed for this object as a function of frequency is consistent with an underlying narrow-band emission process, where the wide-band nature of the measured FRB spectrum is due to relativistic motions. Although this scenario and the simple dynamics we consider could be applied to other theories, they are well-suited for a model based upon Dicke's superradiance as the physical process responsible for FRB radiation in this and similar sources.