This paper focuses on the capacity of the underwater acoustic communications (UAC) channel under realistic assumptions: time-varying multi-paths channel, modelled as a stochastic doubly-spread frequency-selective channel, unknown actual realization of the channel at both transmitter and receiver, constraints on both transmitted power (rms and/or peak) and available frequency bandwidth. The exact channel capacity under such assumptions is still unknown. Therefore, several bounds for this capacity are given and then numerically assessed for a few typical shallow water UAC channels. The main conclusion is that, even with the above comprehensive set of demanding assumptions, as far as the theoretical channel capacity is considered, transmission with higher than often now spectral efficiency (e.g. 2 to 4 bits/sec/Hz) appears as a reasonable objective in typical time-varying underwater acoustic communication channels, providing SNR about 15 to 20 dB.