Underwater acoustic (UWA) communication plays a key role in the process of exploring and studying the ocean. In this paper, a modified non-stationary wideband channel model for UWA communication in shallow water scenarios is proposed. In this geometry-based stochastic model (GBSM), multiple motion effects, time-varying angles, distances, clusters' locations with the channel geometry, and the ultra-wideband property are considered, which makes the proposed model more realistic and capable of supporting long time/distance simulations. Some key statistical properties are investigated, including temporal autocorrelation function (ACF), power delay profile (PDP), average delay, and root mean square (RMS) delay spread. The impacts of multiple motion factors on temporal ACFs are analyzed. Simulation results show that the proposed model can mimic the non-stationarity of UWA channels. Finally, the proposed model is validated with measurement data.