Most rivers exhibit a tendency for the characteristic size of'the bed material to become finer in the downstream direction. In addition, most river sediments also exhibit a paucity of material in the pea gravel range. Because of this paucity the transition in the downstream direction from a gravel-bed stream to a sand-bed stream is usually rather abrupt, and is often marked by a discontinuity in bed slope and stream morphology as well. If the front marking the gravel-sand transition is not to prograde continuously in the direction of the point at which base level is established, e.g. the ocean, then some mechanism must operate to arrest it in place. Here two such mechanisms are examined; abrasion of gravel and basin subsidence (or alternatively base level rise). It is found that either one or a combination of the two can act to stabilize the spatial location of the gravel-sand transition. The present paper is devoted to a simplified analytical solution to the problem that renders the structure of the formulation relatively transparent. A complete numerical solution that more realistically describes the field manifestation is presented in a companion paper.