Ship-borne waves are of significant interest for the design of port and waterway infrastructure and the maintenance of its surrounding environment. Computation of these nonlinear and dispersive waves has mainly been focusing on their near-field generation as a fluid-body interaction problem. This study presents an approach for the computation of ship waves generated by a moving pressure disturbance with phase-resolving and depth-averaged equations. To support a wide range of applicability, the paper deals with the evolution of the vessel wedge compared to an analytical solution for sub-to supercritical speeds and the assessment of wave patterns from a broad range of pressure term dimensions, including cross-references to findings in other studies. The conducted numerical experiments showcase the typical response of a Boussinesq-type model to a simplified moving pressure disturbance and identify the main factors and criteria for ship-wave propagation in the far-field of a vessel. Finally, a unique field dataset underlines the capability of an extended Boussinesq-type model to compute the propagation of vessel waves over an irregular bathymetry.