The transient response of a semi-infinite, thin cylindrical elastic shell to an impulsive, axisymmetric radial excitation applied at the origin is investigated in the time and frequency domains using Fourier and Laplace transform methods. Solutions for the radial and axial displacements as functions of time and axial position show the dispersive nature of the wave propagation. High-frequency components propagating at the plate wave speed are followed by slower components occurring near the shell ring frequency. The temporal variation of the spectral components of the transient displacement is examined using a joint time--frequency distribution function. Analytical results are confirmed by experimental measurements on a brass tube.