Diffraction by a Circular Cavity

The reduced wave equation Δu + k2n2u = 0 is considered, where n = 1 in the exterior of a circular cylinder of radius a and n = N < 1 in the interior. The solution and its first derivatives are required to be continuous for r = a. The behavior of the solution for large ka is described, especially in the vicinity of a ray incident upon the cylinder at the critical angle. The principal novelty of the present results is a description of the solution in certain regions by a combination of geometrical optics (ray contributions) and whispering gallery modes (a kind of diffracted modes). These results are in qualitative agreement with those of Chen, but certain details in Chen's treatment require elaboration. The present results do not agree with those of Nussenzveig for N < 1, especially in the geometrical interpretation. This revision of Nussenzveig's work brings it into agreement with Keller's geometrical theory of diffraction. In fact, part of the present work can be combined with Chen's results to give a fairly complete geometrical theory for diffraction by a convex, transparent object. However, the present treatment does not give a uniform asymptotic expansion of the solution, since the expansion is not given in a transition region in the immediate vicinity of the critically reflected ray.