The usual predictions of linear coupling theory for lower hybrid waves are altered by including an overdense edge plasma (ω < ωpe) in the plasma model. Coupling is found to depend strongly on the value of the edge density, as well as the density gradient. The regimes, where one or the other of these parameters is important, are investigated. Typically, only the first few millimetres of the edge plasma are important in determining coupling. The major implications of the problem of coupling to an overdense plasma can be derived from a simple impedance-matching argument. In general, coupling is optimum for an edge density, n0, determined by , where nc = ω2me/4πe2 and n|| is the parallel index of refraction of the lower hybrid wave.