DYADIC SCATTERING BY SMALL OBSTACLES. THE RIGID SPHERE by GEORGE DASSIOS and KATERINA KARVELI (Division of Applied Mathematics, Department of Chemical Engineering, University of Patras and ICEHT/FORTH, GR 26500 Patras, Greece) [Received 20 January 2000. Revise 14 August 2000] Summary The general theory of low-frequency dyadic scattering is developed for the near fields, the far fields and all the energy functionals associated with scattering problems. The incident field could be any complete dyadic field generated either in the exterior medium of propagation (point source) or at infinity (plane waves). The case of a small rigid sphere, which is illuminated by a plane dyadic field, is solved and the corresponding results for acoustic and elastic scattering are recovered as special cases. In order to solve analytically the sphere problem a special technique had to be developed, which generates Papkovich-type differential representations of dyadic elastostatic displacements. Comparison of numerical results, obtained via the boundary element method, show an amazing accuracy with our analytical results. 1. Introduction Scattering theory deals with the disturbance that a given obstacle causes upon the propagation of a known wave field. Many methods, both analytical and numerical, have been used in order to produce closed-form solutions for the scattering problem. Rayleigh (1 to 4) was the first who considered the problem of approximating solutions of the wave equation in terms of solutions of Laplace’s equation, when the wavelength is large compared to the characteristic dimensions of the scatterer. The general theory of scattering of elastic waves is well exposed by Kupradze (5), where expressions of the fundamental solutions of the spectral Navier equation, integral representations, as well as radiation conditions for the displacement field can be found. Based on Kupradze’s work, Barratt and Collins (6) provided expressions for the scattering amplitudes as well as for the scattering cross-sections in elasticity. Integral representations for the displacement field are also given by Banauch (7) as well as by Wheller and Sternberg (8) who also proved uniqueness theorems. Ying and Truell (9) have investigated the scattering of a longitudinal wave by a sphere while the corresponding problem for a transverse incident wave was solved by Einspruch et al. (10). Dassios and Kiriaki (11) provided a systematic theory for scattering of elastic waves at low frequencies and they applied their theory to the elastic scattering problems by a small rigid ellipsoid (12) and by a small ellipsoidal cavity (13). Based on this theory Kiriaki and Polyzos achieved solutions for the case of a penetrable scatterer with an impenetrable core (14). The case of coated spherical obstacles has been discussed by Kiriaki et al.(15). The goal of the work at hand is to extend the above method to dyadic scattering problems. The displacement fields in this case are complete three-dimensional dyadic fields (16). The advantage of Q. Jl Mech. Appl. Math. (2001) 54 (3), 341–374 c  Oxford University Press 2001