Free alternate bars are large-scale, downstream-migrating bedforms characterized by an alternating sequence of three-dimensional depositional fronts and scour holes that frequently develop in rivers as the result of an intrinsic instability of the erodible bed. Theoretical models based on two-dimensional shallow water and Exner equations have been successfully employed to capture the bar instability phenomenon, and to estimate bar properties such as height, wavelength and migration rate. However, the mathematical complexity of the problem hampered the understanding of the key physical mechanisms that sustain bar formation. To fill this gap, we considered a simplified version of the equations, based on neglecting the deformation of the free surface, which allows us to: (i) provide the first complete explanation of the bar formation mechanism as the result of a simple bond between variations of the water weight and flow acceleration; (ii) derive a simplified, physically based formula for predicting bar formation in a river reach, depending on channel width-to-depth ratio, Shields number and relative submergence. Comparison with an unprecedented large set of laboratory experiments reveals that our simplified formula appropriately predicts alternate bar formation in a wide range of conditions. Noteworthy, the hypothesis of negligible free surface effect also implies that bar formation is fully independent of the Froude number. We show that this intriguing property is intimately related to the three-dimensional nature of river bars, which allows for a gentle lateral deviation of the flow without significant deformation of the water surface.