The nature of stationary-wave structures in multi-ion plasmas with differential streaming between the ions is elucidated by making use of the concept of a generalized sonic point for such plasmas and the Bernoulli equation for each flowing ion species. It is shown that the Bohm criterion for the existence of solitons and double layers is an expression of either the supersonic nature of the collective ion flow or the subsonic nature of individual ion flows. From the perspective of the dispersion equation for stationary waves, both of these requirements, and hence the Bohm criterion, merely ensure that the wave is evanescent, rather than sinusoidal, which, in turn, ensures that the system can evolve nonlinearly. The structure of the ion flows in these transitions is determined by the Bernoulli equation, a fundamental property of which is that the ion plasma energy exhibits a minimum where the ion flow speed equals the local ion sound speed. This implies that for positive flowing ions, a positive change in electric potential induces a deceleration (pile-up) if the flow is supersonic or an acceleration (spread-out) if the flow is subsonic. This feature limits the maximum strength of the potential in a soliton, and sheds light on the underlying reason why double layers can only exist in very special circumstances without the introduction of additional trapped or reflected particle populations.