Transonic solutions of a three-fluid system without cosmic ray diffusion. The transonic solution is subsonic-supersonic transition. In this figure, γ_g = 5∕3, γ_c = 4∕3, GM = 1, a = 1. Left column:pure thermal outflow with cooling (δ_c = 0.1). Middle column: three-fluid outflow with cooling (δ_c = 0.1) but without wave damping (δ⁺ = 0). Right column: three-fluid outflow with cooling (δ_c = 0.1) and wave damping (δ⁺ = 0.1). Upper row: black line and thin brown lines show the flow speed and the effective sound speed of the system. For the three-fluid cases, Alfvén speed is also shown (thin gray lines). Lower row: blue, orange, and green lines show the pressures of the thermal gas, cosmic ray, and forward-propagating wave, respectively. The boundary conditions (P_gb, ρ_b, B_b) are the same for all three cases, and are the same for the two three-fluid cases. In this example, P_gb = 1, P_cb = 0.5, , ρ_b = 1, . The boundary condition U_b is adjusted to obtain the transonic solutions.