Complete families and Rayleigh obstacles

Complete families in a given function space are sets of linearly independen t functions, a linear combination of which can approximate any other functi on 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 obstac le, the obstacle is said to be in the Rayleigh class. One shall further dis tinguish between Dirichlet-Rayleigh and Neumann-Rayleigh obstacles, accordi ng to the applicable boundary condition. Discs are trivial obstacles of the se classes. Ellipses of eccentricity eta such that eta (2) < 1/2 were shown to be in the Dirichlet-Rayleigh class by Barantsev et al. in 1971, who use d the saddle point method to asymptotically estimate the Fourier scattering coefficients. Herewith, another one parameter family of obstacles is const ructed 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.