On the validity of the geometrical theory of diffraction by convex cylinders

In this paper we consider the scattering of a wave from an infinite line source by an infinitely long cylinder C. The line source is parallel to the axis of C, and the cross section C of this cylinder is smooth, closed and convex. C is formed by joining a pair of smooth convex arcs to a circle C₀, one on the illuminated side, and one on the dark side, so that C is circular near the points of diffraction. By a rigorous argument we establish the asymptotic behavior of the field at high frequencies, in a certain portion of the shadow S that is determined by the geometry of C in S. The leading term of our asymptotic expansion is the field predicted by the geometrical theory of diffraction. Previous authors have derived asymptotic expansions in the shadow regions of convex bodies in special cases where separation of variables is possible. Others, who have considered more general shapes, have only been able to obtain bounds on the field in the shadow. In contrast our result is believed to be the first rigorous asymptotic solution in the shadow of a nonseparable boundary, whose shape is frequency independent.