These notes accompany a short student talk entitled ''Searching for light dark matter with radio waves'' and summarise the derivation used in the main work (Zatini et al., 2025) to model resonant axion--photon conversion in isotropic, unmagnetised plasmas. They focus on the theoretical ingredients underlying the conversion mechanism exploited in radio-based searches, rather than on the observational aspects. The key ingredient is the local mass-matching condition $\omega_{\rm pl}(s)\simeq m_a$ (with $\omega_{\rm pl}$ the plasma frequency and $m_a$ the axion mass), which enhances the sourced electromagnetic response. Starting from axion electrodynamics and a cold-plasma dielectric function, we reduce the full 3D problem to an effective 1D description along locally radial trajectories, and obtain the standard stationary-phase/WKB expression for the conversion probability.