The dynamic behavior of a bunched one-dimensional crystalline beam is studied theoretically. It is shown that, owing to the existence of momentum dispersion, a Coulomb chain traveling in a storage ring performs a complex periodic oscillation whenever it is exposed to a longitudinal radio-frequency force. The equations of motion are derived to predict the oscillation pattern in an arbitrary lattice structure. The validity of the present theory is confirmed through multiparticle simulations. Various features of an oscillating string beam, such as the lattice-parameter dependence of the orbit, the stability, and critical line density, etc., are also discussed.