A full understanding of Doppler, either acoustic or electromagnetic, depends on representing the effect in the time domain as well as in the frequency domain. In the time domain, the Doppler perturbation (f - f0)/f0 for a signal becomes its time-delay time derivative. A surprising physical consequence (deduced here for an acoustic reflection problem) is that the Doppler perturbation depends on ratios of distances (source-to-echoer and echoer-to-receiver) as well as on various velocities and angles. Another surprise from the time domain concerns the reverse Doppler effect observed both for electromagnetic and acoustic signals. In certain artificial environments (metamaterials with negative refractive index), radiation can be received with a lower frequency when the source approaches a receiver and with a higher frequency when receding from it. The time-delay time derivative picture represents this effect as a paradox: The time delay (hence source-to-receiver distance) increases when its time-delay time derivative is negative. The restriction of reverse Doppler to a dissipative domain may ameliorate the paradox but does not solve it.