The paper presents an analysis of dynamic stability of long cylindricalsandwich shells and shallow panels subject to a uniform periodic lateral pressure.The solution is obtained using the Sanders shell theory by assumption that the shellor panel remains in the state of plane strain during both steady-state and perturbedmotion. The steady-state motion of a shell is axisymmetric, while perturbedvibrations superimposed on the steady response are asymmetric. The analysis ofperturbed motion is reduced to specifying the conditions of stability of the Mathieuequation. Subsequently, the criteria of dynamic stability and the boundaries of theregions of unstable motion in the pressure amplitude-pressure frequency planeare immediately available. A shallow panel subjected to hydrodynamic pressureexperiences forced vibrations. However, these vibrations can become unstable.Dynamic stability of such vibrations is investigated through the solution of thelinearized equations for perturbed motion. It is shown that these equations can bereduced to a system of Mathieu equations.