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Computer programs have been developed to investigate the construction of polynomial approximations to H-modes in empty guide of arbitrary shape. Mathematically this corresponds to the approximate solution of the Helmholtz equation with homogeneous Neumann boundary conditions. The method used is the well-known Rayleigh-Ritz. Both polynomial and piecewise polynomial function spaces have been investigated for several types of waveguide cross section. Particularly good results have been obtained for convexguides.A comprehensive set of H-mode contour plots are presented for four well-known guide sections.