Summary: We derive one-dimensional wave equations for axially symmetric waves in elastic rods. By using series expansions in the radial coordinate, we obtain a hierarchy of wave equations. As the lowest reasonable approximation, the usual simple wave equation for the rod is recovered. At the next level, we derive a fourth-order wave equation. The dispersion relation and displacements for these approximations and for Love’s equation are compared with the lowest branch of the exact Pochhammer-Chree equation. We also consider an excitation problem with a shear force, and compare the results with other theories.