Real (ω) and imaginary (γ) frequency of the Alfvén wave and its transverse polarisation for different electron-to-proton temperature ratios, with β_p = 0.1 (top) and β_p = 1.0 (bottom). The dispersion relation is plotted with continuous lines for |γ|< |ω| and in dotted lines for |γ|> |ω|, indicating that the mode is strongly damped. Dash-dotted lines indicate k_⊥ρ = 1 for the different configurations, and the segmented line in the real frequency panels corresponds to the MHD dispersion relation of the Alfvén wave.