During somitogenesis in embryos, a posteriorly moving differentiation front arrests the oscillations of "segmentation clock" genes, leaving behind a frozen, periodic pattern of expression stripes. Both mathematical theories and experimental observations have invoked a "clock and wavefront" model to explain this phenomenon, in which long-range molecular gradients control the movement of the front and therefore the placement of the stripes in the embryo. Here, we develop a fundamentally different model-a progressive oscillatory reaction-diffusion (PORD) system driven by short-range interactions. In this model, posterior movement of the front is a local, emergent phenomenon that, in contrast to the clock and wavefront model, is not controlled by global positional information. The PORD model explains important features of somitogenesis, such as size regulation, that previous reaction-diffusion models could not explain. Moreover, the PORD and clock and wavefront models make different predictions about the results of FGF-inhibition and tissue-cutting experiments, and we demonstrate that the results of these experiments favor the PORD model.