Real and imaginary frequency, transverse electrical polarisation, and magnetic polarisation of the Alfvén wave. The summed β of the plasma, β = β_e + β_p, was fixed at 0.12, 0.60, 1.20, and for each value of β different electron-to-proton temperature ratios were considered. Dash-dotted lines indicate k_⊥ρ_L = 1 for the different values of β_e + β_p. The segmented line in the real frequency panels corresponds to the MHD dispersion relation of the Alfvén wave. In the two middle panels it indicates the line corresponding to zero, and in the bottom panel it represents the improved two-fluid approximation of this quantity given in Eq. (13b).