The Green's function for elastic waves is developed. The elastic waves generated by a source contained in a cavity in an infinite elastic medium are written as an integral over the surface of the cavity; the integrand contains the product of the Green's function and the stresses and displacements on the cavity surface. A multipole expansion of this integral is made for sources small compared with the wavelengths radiated. In this case, the first two terms in the expansion accurately describe the radiation. Thus, the explicit time-dependent solutions of the propagation equations are obtained in terms of a few parameters that are calculated by taking prescribed averages of the stresses and displacements on the source surface. The procedure is illustrated by finding the radiation from a simple spherical source.Finally, a variational principle is developed by means of which the displacement on the source surface can be found if only the applied stresses are known.