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An analytic model of the current melt wave in a one-dimensional stationary conductor has been developed to gain insight into the complex problem of melt-wave erosion contact wear in railgun armatures. Unlike two-dimensional models with motion, in which the dominant driving mechanism for the erosion front is current concentration from the velocity skin effect, in one dimension, the driving mechanism is a concentration of current caused by the electrodynamics at the melt-wave erosion boundary. Specifically, as molten material is ejected, current is driven into the remaining solid material at the interface, resulting in a local concentration of current and joule heating at the interface. We derive an expression for the velocity of the melt-wave front in one dimension, and assess the importance of this effect by comparing the erosion speed we obtain with erosion speeds predicted by previous models. The comparison suggests that the electrodynamics of the moving melt-wave boundary has an insignificant effect on melt-wave erosion in solid armature railguns, and as such can be justifiably neglected.