The effects of the ion temperature are discussed in a two-ion electron plasma and for a model applicable to the oscillating sheath theory that has recently been much of the focus in research. The differences between the fluid and kinetic models are pointed out, as well as the differences between the approximative kinetic description (which involves the expansion of the plasma dispersion function), and the exact kinetic description. It is shown that the approximative kinetic description, first, cannot describe the additional acoustic mode which naturally exists in the plasma with an additional ion population with a finite temperature, and, second, it yields an inaccurate Landau damping of the bulk ion acoustic mode. The reasons for these two failures of the model are described. In addition to this, a fluid model is presented that is capable of capturing both of these features that are missing in the approximative kinetic description, i.e., two (fast and slow) ion acoustic modes, and the corresponding Landau damping of both modes.