Wave splittings are derived for three types of structural elements: membranes, Timoshenko beams, and Mindlin plates. The Timoshenko beam equation and the Mindlin plate equation are inherently dispersive, as is each Fourier component of the membrane equation in an angular decomposition of the eld. The distinctive feature of the wave splittings derived in the present paper is that, in homogeneous regions, they transform the dispersive wave equations into simple one-way wave equations without dispersion. Such splittings have uses bothfor radial scattering problems in the 2D cases and for scattering problems in dispersive media. As an example of how the splittings may be applied, a direct scattering problem is solved for a membrane with radially varying density. The imbedding method is utilised, and agreement is obtained with an FE simulation.