Dispersion relation for circularly polarised Alfvén waves calculated using Eqs. (48) and (49). The right-handed mode or R-mode has a frequency that is unbounded with the wavenumber k. The left-handed mode or L-mode in the massless electron model considered here has a frequency that tends asymptotically for large k to the ion-cyclotron frequency, ω_cp. The dashed line represents the results of the single-fluid model and in this case the Alfvén wave has a frequency given by ω = kv_A. In the limit of very small numbers the wave frequencies of the single- and multi-fluid descriptions coincide.