This paper presents a coupled finite element and boundary integral method for solving the time-periodic oscillation and scattering problem of an inhomogeneous elastic body immersed in a compressible, inviscid, homogeneous fluid. By using integral representations for the solution in the infinite exterior region occupied by the fluid, the problem is reduced to one defined only over the finite region occupied by the solid, with associated nonlocal boundary conditions. This problem is then given a family of variational formulations, including a symmetric one, which are used to derive finite-dimensional Galerkin approximations. The validity of the method is established explicitly, and results of an error analysis are discussed, showing optimal convergence to a classical solution.