In the present paper we adapt the transition matrix method to the case of a smooth elastic obstacle. The boundary conditions employed are discussed in some detail and are seen to be responsible for the fact that the rather complicated formal solution in many respects resembles that for an elastic obstacle in a fluid. Comparisons are made between the smooth and the usually employed welded obstacles, and for spheres approximate expressions are given for the transition matrices in the long wavelength limit. Numerical results (cross sections and surface stresses) are given for spheres and spheroids, and it is then seen that often the smooth and welded obstacles, at least qualitatively, show similar features.