A new system of equations is derived for compressible shallow-water flows with a depth-averaging method and a weak-compressibility assumption. The variations of the depth-averaged density are due to the variations of the hydrostatic pressure caused by depth variations. The obtained system of four equations is fully nonlinear, hyperbolic and admits an exact conservation of energy on an arbitrary bathymetry as well as on a mild-slope bottom. The dispersive properties are consistent with the linear theory of compressible fluids at the long-wave limit. The equations include the possibility of a mobile bottom, enabling the simulation of tsunamis generated by seabed vertical movements. A system with improved dispersive properties is presented with accurate velocities for all tsunamis wavelengths. The solutions of the dispersive relation divide into a slow branch governing gravity waves and a rapid branch governing acoustic waves. The numerical scheme is based on a splitting between a slow part treated explicitly and a fast part solved implicitly but without any global linear system to solve. The numerical resolution of a one-dimensional tsunami generated by vertical bottom movements shows the decrease of the tsunami velocity due to compressible effects and a later arrival time than in an incompressible case.