Alfvén wave frequency vs. normalized wave number. The top (bottom) panel shows the real (imaginary) part of the normalized frequency ϖ = ω/k_zv_A vs. the normalized wavenumber a = k_zv_A/ν_in. The blue line corresponds to the solution of the single-fluid dispersion relation, i.e. Eq. (44) and red asterisks are the solutions of the two-fluid dispersion relation, Eq. (42). The values are calculated for 50% of neutral hydrogen, ξ_n = 0.5.