We study the software simulation of underwater acoustic communications with mobility in UWASNs. We simulate the mobility of sensors and their communications, moving according to the Meandering Current Mobility (MCM) model. OMNeT++ and MATLAB are used to model and integrate three protocol layers of UWASNs: network, link, and physical. The effects of noise and attenuation on the underwater communications are considered in the physical layer model. Three major metrics are calculated in this work: Bit Error Rate (BER), distance, and signal energy per bit over noise power spectral density (Eb/N0). Finally, the performance of communicating sensors moving according to the MCM model is evaluated. The simulation results illustrate that the BER for digital data signals is an increasing function of the distance. In addition, the BER for digital data signals is a decreasing function of the Eb/N0 ratio.