The differential equation for radial vibrations of a thin cylindrical shell is derived by Hamilton's principle for the case of constant internal and transient external pressures. Both pressures are assumed symmetrical about the cylinder's axis of symmetry, and the natural frequencies and mode shapes are obtained. The general solution to the transient forced‐vibration problem is presented using integral transform methods. The general solution is then carried out in detail for the specific case where the external pressure is a step function moving over the cylinder in the axial direction. It is pointed out that such a solution may be considered as a crude model of what happens to a thin‐skinned missile during launching from a silo. Several other problems are mentioned which have the same type of governing differential equations and for which the solutions presented would be useful.