A new theory of contact-induced standing waves in rotating tyres is presented. The tyre is run by a rotating drum with a fixed shaft distance as in experiments. The tyre belt is modelled as a rotating ring whereas its sidewalls are modelled via radial stiffness of an elastic foundation supporting the ring. The differential equation of vibrations is reduced to the radial deflection of the ring. Critical rotation speed depends on the inflation pressure and the ring bending stiffness. Different tyre response functions are defined in the rotation speed domain with respect to the critical speed. The contact region between the tyre and the drum is defined considering the increase of the tyre radius due to the centrifugal load. Appropriate boundary conditions are specified in order to ensure a continuity of the ring deformation. Bearing reaction forces, as a result of drum penetration into the tyre, are defined.