Discrete methods of numerical analysis have been used successfully for decades for the solution of problems involving wave diffraction, etc. However, these methods, including the finite element and boundary element methods, can require a prohibitively large number of elements as the wavelength becomes progressively shorter. In this paper, a new type of interpolation for the wave field is described in which the usual conventional shape functions are modified by the inclusion of a set of plane waves propagating in multiple directions. Including such a plane wave basis in a boundary element formulation is found in this paper to be highly successful. Results are shown for a variety of scattering/radiating problems from convex and nonconvex obstacles on which are prescribed piecewise constant Robin conditions. Notable results include a conclusion that, using this new formulation, only approximately three degrees of freedom per wavelength are required
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