The potential benefits of massive multiple-input multiple-output (MIMO) make it possible to achieve high-quality underwater acoustic (UWA) communications. Nevertheless, due to the wideband nature of UWA channels, existing massive MIMO techniques for radio frequency cannot be directly applied to UWA communications. This paper investigates a UWA massive MIMO system in the shallow-water environment, deploying large array apertures at both the transmitter and the receiver. We propose a beam-based UWA massive MIMO channel model and analyze its properties. Based on this model, we reveal that the transmit design for rate maximization can be performed in a dimension-reduced space related to the channel taps. Then, we prove that the beam-domain transmission is optimal to maximize the rate when with unlimited numbers of transducers. Furthermore, if the number of hydrophones also tends to infinity, the optimal power allocation can be obtained just by the water-filling algorithm and the corresponding rate positively correlates with the number of channel taps for the high signal-to-noise-ratio regime. Moreover, we devise a low-complexity algorithm to optimize the input covariance matrix for general cases. Simulation results illustrate the significant performance of the proposed algorithm and the high throughput achieved by massive MIMO.