Abstract: The finite element numerical method is used to study dynamic buckling of cylindrical shells under elastic compression waves. The finite element characteristic equations are derived on the basis of the adjacent-equilibrium criterion and the stress wave theory. In these equations, the compression wave propagation and the transverse inertia effect are taken into consideration. By introducing the dynamic buckling supplementary restraint conditions at the compression wave front of cylindrical shells at the instant when the buckling occurs, the critical-load and dynamic buckling modes of the shells are calculated from the solutions of the finite element characteristic equations. Calculated results provide some insight into the buckling mechanism as a transient process and the influences of the axial stress waves on the buckling deformation.