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Highlights • An original time-split approach is implemented in a free-surface ocean model. • NH flows are simulated in a natural way, propagating acoustic waves. • The three-mode algorithm is detailed and its performances are evaluated. • Acoustic, surface-gravity and internal-gravity waves are studied Abstract A three-mode time-split algorithm is proposed to simulate non-hydrostatic ocean processes in a natural way: fast pressure adjustments via acoustic waves are explicitly represented. The full set of compressible Navier–Stokes equations is integrated for a free-surface ocean with three time-steps respectively associated to the internal, external and compressible (non Boussinesq) modes. The resulting three-mode algorithm can be implemented either in a fully compressible configuration or in a "pseudo-compressible" configuration by artificially lowering acoustic-wave velocities to reduce computational costs. The present description of the three-mode algorithm focuses in the first place on the simulation of non-hydrostatic processes in free-surface ocean models. Several test simulations related to linear and non-linear acoustic, surface and internal gravity waves are studied in details with the proposed three-mode algorithm. [ABSTRACT FROM AUTHOR]