When an underwater acoustic modem is installed on a mobile platform such as an underwater vehicle, a buoy, or a surface vessel, Doppler effects distort the acoustic signal significantly. The acoustic path between a surface vessel and an underwater vehicle, for example, can experience Mach numbers of one percent and more which can be catastrophic if not compensated dynamically. In this paper, we derive a sample-by-sample, recursive resampling technique, in which time-varying Doppler is explicitly modeled, tracked and compensated. Integrated into an iterative turbo equalization based receiver, this novel Doppler compensation technique achieves unprecedented communication performance in field tests and simulations. Our field data stems from the MACE10 experiment conducted in the shallow waters 100 km south of Martha's Vineyard, MA. Under challenging conditions (harsh multi-path, ranges up to 7.2 km, SNRs down to 2 dB and relative speeds up to 3 knots) our receiver achieved a raw data rate of over 39 kbits/s and a perfectly reliable net data rate of over 23 kbits/s (taking into account the overhead from equalizer training and channel coding) in less than 10 kHz of bandwidth. To illustrate the robustness of this approach to high rates of Doppler, a variety of simulations are also provided. We demonstrate that the performance of our algorithm does not depend on the absolute level of Doppler, but only on the rate of its variation.