Diffraction of two dimensional elastic waves by a cylindrical cavity in a half-space is considered in this paper. The paper is divided into two parts. In the first part we use an integral representation for the scattered displacement field to derive its asymptotic behavior at large distances from the cavity. We also give a formal derivation of the representation of the displacement field in terms of multipolar potentials. In the second part we use a method of matched asymptotic expansions to calculate the scattered field. Numerical results are presented for the surface displacements due to Rayleigh waves scattered by a circular cylindrical cavity.