In previous work (Myra J R and D'Ippolito D A 2008 Phys. Rev. Lett.101 195004) we studied the propagation of slow-wave (SW) resonance cones launched parasitically by a fast-wave antenna into a tenuous magnetized plasma. Here we extend the treatment of SW propagation and sheath interaction to 'dense' scrape-off-layer plasmas where the usual cold-plasma SW is evanescent. Using the sheath boundary condition, it is shown that for sufficiently close limiters, the SW couples to a sheath–plasma wave and is no longer evanescent, but radially propagating. A self-consistent calculation of the rf-sheath width yields the resulting sheath voltage in terms of the amplitude of the launched SW, plasma parameters and connection length. The conditions for avoiding potentially deleterious rf-wall interactions in tokamak rf heating experiments are summarized.