A residually stressed thick-walled tube is subject to a combination of internal or external pressure and axial stretch to maintain its circular cylindrical geometry. A small incremental motion is then superimposed on the finitely deformed configuration. We summarize the basic elements of the theory of hyperelasticity with initial stress, together with the corresponding incremental forms. Equations are derived and used to evaluate the dependence of the axial load, of the vibration frequency and of the circumferential stress, on the inner and outer curved boundaries, on the residual stress and on various geometric quantities. The theory is applied to a simple neo-Hookean model with two material parameters and a parameter to scale the magnitude of the residual stress. The static response to small incremental deformations is obtained as a special case when the vibration frequency vanishes.