We consider the problems of propagation of elastic and elastoviscoplastic waves in laminated and fiber composites on the basis of the two-velocity model proposed in [1]. We study plane waves propagating parallel to the fibers and waves propagating in the perpendicular direction. We compare our solutions with the experimental results presented in the literature [2, 3] for several specific composites (carbon-filled plastics, boron plastics, and wolfram fibers in aluminum matrix). The coincidence between the theoretical results and experimental data is shown to be good.