We consider a fast solar wind-type plasma consisting of electrons, a protons background, and a much more tenuous alpha particle beam. It is shown that such system is unstable against ion acoustic-like waves triggered by finite-amplitude left-hand polarized waves propagating in the direction of the interplanetary magnetic field. The ion acoustic-like waves have similar properties to those found in a plasma with a much denser proton beam (Gomberoff, 2006b). Although the alpha particle beam is much more tenuous than the proton beam, the system is equally able to trigger ion acoustic-like waves. Therefore we believe that this effect should be taken into account in the evolution of the fast solar wind.