We investigate the propagation of dissipationless, hydromagnetic, toroidal Alfvén waves in a realistic background, low-latitude fast solar wind with differentially flowing protons and alpha particles. Symmetry about the magnetic axis is assumed. Without invoking the short-wavelength WKB approximation, we derive the equations governing the wave transport from standard five-moment equations. The Alfvénic point, where the combined poloidal Alfvén Mach number MT = 1, is found to be a singular point for the wave equation, which is then numerically solved for three representative angular frequencies W = 10-3, 10-4, and 10-5 rad rad-1, with an amplitude of 10 km s-1 imposed at the coronal base (1 R⊙). Between 1 R⊙ and 1 AU, the numerical solutions show substantial deviation from the WKB expectations. Even for the relatively high frequency W = 10-3 rad S-1, WKB-like behavior can be seen only in re gions r ≲ 10 R⊙, For W = 10-5 rad s -1, the computed profiles of wave-related parameters show a spatial dependence distinct from the WKB one, the deviation being particularly pronounced in interplanetary space. In the inner corona r ≲ 4 R⊙, the computed ion velocity fluctuations and wave-induced acceleration exerted on protons or alpha particles are considerably smaller than their WKB counterparts. With the chosen base wave amplitude, the wave acceleration has negligible effect on the ion force balance in the corona. However, at large distances beyond the Alfvénic point, the low-frequency wave with W = 10 -5 rad s-1 can play an important role in the ion dynamics, with the net effect being to equalize the speeds of the two ion species considered.