We consider a simple one-dimensional model of a rotating balloon tire, with radial displacements, and with only one degree-of-freedom at each circumferential location. This model is motivated by our study of a small balloon tire in the laboratory. Our model has linear material behavior, but the problem is rendered nonlinear due to contact between tire and ground. Analytical results are obtained by explicitly examining steady state solutions to the nonlinear boundary value problem. The results from the model predict standing waves that are qualitatively similar to those observed experimentally. The effects of adding a small beam term and small damping to the dominant membrane terms are discussed. Standing wave wavelengths and spatial attenuation rates are related to tire rotation speed. We discuss the way in which our treatment differs from other studies in the literature which use natural frequencies and mode shapes, or wave propagation ideas.