QUARTERLY OF APPLIED MATHEMATICS VOLUME XLVI, NUMBER 2 JUNE 1988, PAGES 285-299 THE ATKINSON-WILCOX EXPANSION THEOREM FOR ELASTIC WAVES* By GEORGE DASSIOS1 University of Patras, Greece Abstract. Consider the problem of scattering of an elastic wave by a three-dimen- sional bounded and smooth body. In the region exterior to a sphere that includes the scatterer, any solution of Navier's equation that satisfies the Kupradze's radiation condition has a uniformly and absolutely convergent expansion in inverse powers of the radial distance from the center of the sphere. Moreover, the coefficients of the expansion can recurrently be evaluated from the knowledge of the leading coefficient, known as radiation pattern. Therefore, a one-to-one correspondence between the scattered fields and the corresponding radiation patterns is established. The acoustic and electromagnetic cases are recovered as special cases. 1. Introduction. The Sommerfeld's radiation condition [17] imposes, upon any scalar scattered field, the appropriate asymptotic characteristics in order for the scat- tering problem to have a unique solution. It actually states that the behaviour of the scattered field, far away from the scatterer, should coincide with the behaviour of an outgoing spherical wave emanating from an oscillatory point source. Miiller [13] and Silver [16] provided the corresponding radiation condition for electromag- netic scattering, while in the case of elastic wave scattering, the appropriate radiation conditions for the longitudinal as well as the transverse waves are due to Kupradze [8]. Stoker [18], [19] attempted to replace the radiation condition, which is actually imposed upon the spatial behaviour of the time-independent wave solution, with ap- propriate initial conditions imposed upon the (time-dependent) solution of the wave equation, and then find the solution of the time-independent problem by passing to the limit as time tends to infinity. He illustrated his idea by a few examples taken from hydrodynamics. Wilcox [26] put this idea in general framework, by showing that Sommerfeld's radiation condition for a time-harmonic wave is a consequence of the fact that a solution of the initial-boundary value problem represents a wave propagating outward from the source (scatterer) with a constant velocity. His method is based on the concept of spherical means. It was Atkinson [1], in 1949, who first realized that the Sommerfeld's asymptotic condition has an exact counterpart in terms of a convergent series representation *Received December 31, 1986. 'Visiting the Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300. ©1988 Brown University 285