Complete families and Rayleigh obstacles (ICTCA '99)

Complete families in a given function space are sets of linearly independent functions, a linear combination of which can approximate any other function with arbitrarily high accuracy. Outgoing cylindrical wave functions are one such family, used to represent the scattered wave in exterior boundary value problems for the scalar Helmholtz equation in two spatial dimensions. When the incident wave is plane and the scattered wave is represented by a series of said functions, which converges up to the boundary of the obstacle, the obstacle is said to be in the Rayleigh class. One shall further distinguish between Dirichlet-Rayleigh and Neumann-Rayleigh obstacles, according to the applicable boundary condition. Discs are trivial obstacles in these classes. Ellipses of eccentricity $\eta$ such that $\eta ^2 < 1/over 2$ were shown to be in the Dirichlet-Rayleigh class by Barantsev et al. in 1971, who used the saddle point method to asymptotically estimate the Fourier scattering coefficients. Herewith, another one parameter family of obstacles is constructed by the same method. It is also shown that the same obstacles are in the Neumann-Rayleigh class. The relevance of these results to the numerical treatment of scattering problems is briefly discussed.