The Reissner linearized equations of motion, including hysteretic‐ and viscous‐damping terms, are adopted in the study of the dynamics of a finite, thin, elastic cylindrical shell of circular cross section. The integral representation of the radial displacement solution is obtained for an arbitrary, time‐dependent pressure field over the shell surface and the associated Green's‐function solution is calculated. In particular, the shell response to a moving shock wave (step‐function discontinuity), impinging upon the shell at an arbitrary angle of incidence, is calculated. In the special case of axial symmetry, two additional waveforms are considered, that of a moving exponentially decaying shock wave and of a ramp/step‐function wave. Typical numerical results are given in the form of displacement‐time curves and the effects of the pressure‐wave parameters (decay and rise time) are examined. The dynamic behavior of a prepressurized shell is also considered. It is shown that the Green's‐function solution remains form‐invariant under the introduction of an internal static pressureP0; this, in turn, leads to a simple rule for deducing the response to a given external pressure field whenP0≠0 from known solutions corresponding to the case ofP0= 0.