We present an analytic study of the dispersion relation for an isotropic, non-degenerate plasma interacting with an axion-like particle through the axion-photon coupling. We aim to provide a general qualitative picture of the electromagnetic oscillations for both the ultra-relativistic and non-relativistic plasmas, taking into account separate parts of the dispersion curves depending on the plasma temperature and the ratio of the phase velocity to the characteristic velocity of the particles. We include the treatment of the Landau damping, which could be responsible for a sympathetic damping of axions in the plasma waves. We show that there are two kinds of electromagnetic oscillations, i.e. normal super-luminous, undamped and sub-luminous, aperiodically damped oscillations.