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Ideal MHD provides an accurate description of low-frequency Alfv\'en waves infully ionized plasmas. However, higher frequency waves in many plasmas of thesolar atmosphere cannot be correctly described by ideal MHD and a more accuratemodel is required. Here, we study the properties of small-amplitudeincompressible perturbations in both the low and the high frequency ranges inplasmas composed of several ionized species. We use a multi-fluid approach andtake into account the effects of collisions between ions and the inclusion ofHall's term in the induction equation. Through the analysis of thecorresponding dispersion relations and numerical simulations we check that athigh frequencies ions of different species are not as strongly coupled as inthe low frequency limit. Hence, they cannot be treated as a single fluid. Inaddition, elastic collisions between the distinct ionized species are notnegligible for high frequency waves since an appreciable damping is obtained.Furthermore, Coulomb collisions between ions remove the cyclotron resonancesand the strict cut-off regions that are present when collisions are not takeninto account. The implications of these results for the modelling ofhigh-frequency waves in solar plasmas are discussed.

Multi-fluid approach to high-frequency waves in plasmas: I. Small-amplitude regime in fully ionized medium