A simple construction is given, in terms of potentials, for assigning P and S components of elastic-wave displacement in media with depth-dependent density and Lame parameters. Attention is focused on spherically symmetric media, with radial dependence of material properties, and the asymptotic methods used are expected to be accurate for seismic body waves and surface waves with period less than about 1 min. Novel features of the construction include: (a) its completeness (all possible displacements are represented by the potentials), (b) the development of second-order wave equations, for the P and S potentials, which explicitly display a coupling coefficient, and (c) demonstration of the way in which P and SV components of displacement decouple as frequency increases. Previous work has given constraints on the medium such that P and SV decouple completely: these constraints are here simplified and are seen to arise naturally in the context of the present potential representation. The coupling coefficient (between P and SV) is examined in a variety of earth models.