The effect of three factors—the order of fractional derivative, the hydrostatic initial stress, and parameter of magnetic field—on the plane waves in a half-space made of fiber-reinforced material described by the theory of generalized magneto-thermoelasticity is studied. The problem is solved numerically using the normal mode analysis. The results correspond to the Lord–Shulman model and the model that uses fractional derivatives and are represented in the form of graphs. The findings show pronounced effect of the three factors. The results are compared with the case where the initial stress and magnetic field are absent © 2016, Springer Science+Business Media New York.