Contour plots of the first critical fast Mach number in the plasma β - θ ₁ plane for ${E}_{\mathrm{CR}}/{\rho }_{1}{v}_{1| | }^{2}=0.025$ (left panels) and 0.1 (right panels), showing initially forward-propagating waves only (top panels), initially balanced waves (middle panels), and initially backward-propagating waves only (bottom panels).