A representation is proposed for the small oscillations of bodies in unbounded uniformly stratified Boussinesq fluids, in terms of a surface distribution of singularities. The distribution satisfies an integral equation, expressing the continuity of normal velocity. The equation is solved for a sphere and a circular cylinder; when these are rigid, the form of the distribution is interpreted in terms of added mass. In a stratified fluid, owing to the generation of internal waves by vertical motion (propagating waves at frequencies below the buoyancy frequency, evanescent waves at frequencies above it), added mass becomes anisotropic and frequency-dependent. It is only in the limit of large frequencies that the coefficients of added mass in a homogeneous fluid, 1/2 for the sphere and 1 for the circular cylinder, are recovered. Two definitions of added mass are considered, based on the impulse of the fluid and on the pressure on the body, respectively. In a homogeneous fluid they are equivalent; stratification makes them distinct, leading to two different forms of added mass. Two applications are considered: the wave power radiated by the forced oscillations of a body; and the free oscillations of a body displaced from its equilibrium level then released. A maximum of the wave power is observed at a practically constant fraction 0.8 of the buoyancy frequency, and its implications for the radiation of waves from regions of random fluid motion are discussed.