A f o r m u l a t i o n of e l a s t o d y n a m i c diffraction problems for sinusoidally in time v a r y i n g d i s t u r b a n c e s in a linearly elastic m e d i u m is presented. S t a r t i n g w i t h t h e e l a s t o d y n a m i c reciprocity relation, a n integral r e p r e s e n t a t i o n for t h e particle d i s p l a c e m e n t is derived. I n it, t h e particle d i s p l a c e m e n t a n d t h e t r a c t i o n at t h e b o u n d a r y of t h e obstacle occur. F r o m t h e integral represent a t i o n , a n associated integral e q u a t i o n is o b t a i n e d b y letting t h e p o i n t of o b s e r v a t i o n a p p r o a c h t h e b o u n d a r y of t h e obstacle. The " o b s t a c l e " m a y be either a rigid body, a void, or a b o d y w i t h elastic properties differing from those of its e n v i r o n m e n t , or a c o m b i n a t i o n of these. T h e integral e q u a t i o n t h u s o b t a i n e d is well-suited for n u m e r i c a l t r e a t m e n t , w h e n obstacles up to a few w a v e l e n g t h s in m a x i m u m d i a m e t e r are considered. Β§ 1. Introduction Many existing approaches to solving the problem of the diffraction and scattering of acoustic waves by an obstacle in an elastic medium are based upon the technique of separation of variables and a corresponding expansion of the unknown vectorial particle displacement in a series of associated transcendental functions. In this category, White ~13 considers the scattering, of both compressional and shear waves, impinging obliquely on an infinitely long cylinder of circular cross-section by expanding the unknown particle displacement in a series of Hankel and Bessel functions. The "obstacle" can be either a void, a rigid inclusion, or a cylinder with elastic properties differing from those of its environment. Also, numerical * The research reported in this paper has been supported by the Netherlands organization for the advancement of pure research (Z.W.O.).