Plasma-frame wave frequency ω vs. wavenumber k from the cold plasma approximation, with ω normalized to the proton gyrofrequency ω_p and k to ﹩{d}_{p}^{-1}﹩. Panel (a) shows the upstream interval, and panel (b) shows the downstream interval. A/IC (red) and FM/W (blue) modes are shown; solid lines show the proton–electron cold plasma dispersion relation, while dashed lines include alpha particles. Intersections with the Doppler-shift-derived relation (orange) indicate possible wavevector solutions (black circles).