We have developed a demodulator for low data rate, asynchronous frame, and narrow bandwidth underwater acoustic communication. We aim at operation under harsh conditions, ie, low signal-to-noise ratio, and across long distances. In this paper, we pay a special attention to the efficiency of mobility support. Mobility results into the Doppler effect, which, for a demodulator, makes the carrier frequency drift arbitrarily during attempts to decode frames. The chances of success are better when the demodulator can tune into the drifted carrier frequency. This can be achieved by trying a range of possible drifted carriers. We introduce the novel idea of normalized trajectory. Each normalized trajectory produces a unique Doppler shift pattern that can be applied to tune into a drifted carrier. We demonstrate that this improvement is theoretically sound. From a practical point of view, the search space is potentially reduced. The actual gain in performance is application-specific and depends on the actual sets of trajectory parameters that are considered. We introduce the concept of normalized trajectory, discuss its integration into the demodulator, and review the performance of the new design.We have developed a demodulator for low data rate, asynchronous frame, and narrow bandwidth underwater acoustic communication. We aim at operation under harsh conditions, ie, low signal-to-noise ratio, and across long distances. In this paper, we pay a special attention to the efficiency of mobility support. Mobility results into the Doppler effect, which, for a demodulator, makes the carrier frequency drift arbitrarily during attempts to decode frames. The chances of success are better when the demodulator can tune into the drifted carrier frequency. This can be achieved by trying a range of possible drifted carriers. We introduce the novel idea of normalized trajectory. Each normalized trajectory produces a unique Doppler shift pattern that can be applied to tune into a drifted carrier. We demonstrate that this improvement is theoretically sound. From a practical point of view, the search space is potentially reduced. The actual gain in performance is application-specific and depends on the actual sets of trajectory parameters that are considered. We introduce the concept of normalized trajectory, discuss its integration into the demodulator, and review the performance of the new design.