Abstract :
The famous Atkinson–Wilcox theorem claims that any scattered field, no matter what the
boundary conditions on the surface of the scatterer are, can be expanded into a uniformly
and absolutely convergent series in inverse powers of distance and that once the leading
coefficient of the expansion is known the full series can be recovered up to the smallest
sphere containing the scatterer in its interior. The leading coefficient of the series is nothing
else but the scattering amplitude. This is a very useful theorem, which provides the exact
analogue of the Sommerfeld radiation condition, but it has the disadvantage of recovering
the scattered field only outside the sphere circumscribing the scatterer. This means that an
elongated obstacle which has a very large, as it compares to its volume, circumscribing
sphere leaves a lot of exterior space where the scattered field cannot be recovered from its
scattering amplitude. In the present work the Atkinson–Wilcox theorem has been extended
to the ellipsoidal system where the theorem as well as the relative recovering algorithm
holds true all the way down to the smallest circumscribing ellipsoid. Considering the
anisotropic character of the ellipsoidal geometry it is obvious that an appropriately chosen
ellipsoid can fit almost every smooth convex obstacle. Furthermore, such a result offers
the best opportunity to develop a hybrid method based on the theory of infinite elements.
Two orientations dependent differential operators are introduced in the recurrence scheme
which, as the ellipsoid degenerates to a sphere, one of them vanishes, while the other
reduces to the Beltrami operator. A reduction to spherical geometry is also included.
2002 Elsevier Science (USA). All rights reserved.
