In this paper we analyze the performance of adaptive orthogonal frequency division multiplexing (OFDM) for underwater acoustic (UWA) communications. Our goal is to maximize the system throughput under a target average bit error rate (BER). We consider two different schemes based on the level of adaptivity: in the first scheme only the modulation levels are adjusted while the power is allocated uniformly across the sub-carriers, whereas in the second scheme, both the modulation levels and the power are adjusted adaptively. We exploit a feedback from the receiver in two forms: one that conveys the modulation alphabet and quantized power levels to be used for each sub-carrier, and another that conveys a quantized estimate of the sparse channel impulse response. We compare these two strategies based on the average number of bits per OFDM block needed to achieve a pre-specified target average BER. In both cases we predict the channel one travel time ahead so as to improve the performance in the presence of a long propagation delay. The second approach is shown to be advantageous, as it requires significantly fewer feedback bits for the same system throughput. The effectiveness of the proposed schemes is demonstrated using real channel measurements recorded in shallow water off the western coast of Kauai, Hawaii, in June 2008.