Axially symmetric waves in semi‐infinite hollow elastic circular cylinders, with traction‐free lateral surfaces and subjected to a pressure‐step loading, are considered on the basis of the equations of motion derived by McNiven, Shah, and Sackman [J. Acoust. Soc. Amer.40, 784–792; 1073–1076 (1966)]. By using double integral transforms, asymptotic solutions are obtained for the strains as functions of time and material and geometric parameters. Strains are presented for times near the head of the pulse of the extensional mode at the same distance from the finite end for various radii ratiosa* and for different distances but for the same radius ratio. It is of interest to note that, for thin cylinders (a* less than approximately 1.7, owing to its slight dependence on Poisson's ratio), the maximum group velocity of the radial mode is slightly higher than that of the extensional mode although less than the dilatational wave velocity.