We report a novel method to continuously track the temporal evolution of an arbitrary complex waveform as it propagates through a group-velocity dispersion medium by using a single-frequency-driven phase modulator. The proposed method exploits the fact that the frequency spectrum of a given (input) waveform, following a suitable sinusoidal temporal phase modulation, exhibits the same shape as that of a dispersed version of the same temporal waveform after propagation through a prescribed amount of dispersion. In experiments, we track the dispersion-induced temporal evolution of different optical picosecond pulsed waveforms by tuning the frequency and/or amplitude of the phase modulation signal and observing the resulting shapes in the optical frequency domain. A good agreement is obtained between the measured spectra and predicted temporal shapes of the propagating waveform for different amounts of dispersion. Moreover, the method is successfully applied on a chirped optical pulse to find the optimal pulse compression conditions.