Abstract: Solutions to the equation of waves motion are derived for homogeneous and transversely isotropic media such as fiber-reinforced composites, and three dimensional slowness surfaces are shown as well. A brief discussion on the propagation of plane waves is given. Elastic plane waves are characterized by slowness vectors, wave vectors, polarization vectors and group velocity vectors, etc. The results obtained are presented in a coordinate-free form due to the introduction of the crystal axis' orientation vector which specifies the anisotropy of the media. Therefore, the results are the most general and convenient for further application.