Example of transonic and subsonic solutions of a four-fluid system with both forward- and backward-propagating waves. In this figure, γ_g = 5∕3, γ_c = 4∕3, GM = 1, a = 1, η = 1. Two cases are compared, (1) dashed line: cooling is “OFF” (δ_c = 0) and (2) solid line: cooling is “ON” (δ_c = 0.1). Wave damping is “ON” for both cases (δ^± = 1). Left panel: black, thin brown, and grey lines show the flow speed, the effective sound speed, and the Alfvén speed, respectively. Right panel: blue, orange, green, and purple lines show the pressures of thermal gas, cosmic ray, forward-propagating wave, and backward-propagating wave, respectively. The boundary conditions are the same for the two cases. In this example, U_b = 2∕3, P_gb = 1, P_cb = 1, , , ρ_b = 1, . The cosmic ray pressure gradient or diffusive flux at the base is adjusted to get the transonic solution if cooling is “ON” and get a physically allowable subsonic solution if the cooling is “OFF.”