The response of a semiinfinite transversely isotropic rod to a time‐dependent input on the end of the rod is found using the method of characteristics. The rod is of circular section and the material of the rod is arranged so that the axes of isotropy are parallel to the axis of the rod. To reduce the problem to one where motions depend on only one space variable and time, use is made of an approximate theory developed by the authors in a previous study. Numerical results are found for an input which has a step distribution in time and for two different materials, magnesium and a fiber‐reinforced material. For each material the response is found in terms of four quantities: radial strain, axial strain both on the lateral surface and along the axis of the rod, and the generalized axial stress.