Wave frequency as a function of wavelengths for n₀ = 10⁶ cm⁻³ and a single grain size of a = 3 × 10⁻⁷ cm. In the top panel, the grain inertia is considered and the three modes are displayed (dark, blue, and red lines). Both the real (solid lines) and the imaginary (dashed lines) parts are shown. In the bottom panels, the grain inertia is neglected and the two corresponding modes are displayed (cyan and purple). The modes of top panel have been drawn in the bottom panel (thin lines) for ease of comparison. At large wavelengths, inertia does not play any role. However at a scale smaller than about 10 au, the inertia drastically modify the Alfvén wave propagation. The yellow, green, pink, and orange vertical lines represent λ_{AW, diss, n}, λ_{AW, diss, d}, λ_{AW, hall, n} and, λ_{AW, hall, d}, respectively (see Eqs. (41)–(44)). We note that in this particular case, λ_{AW, diss, d} = λ_{AW, hall, n} and so, the pink line is not visible.