In the present work, the modal fields and cutoff wavenumbers of arbitrary cross section waveguides are determined using the Rayleigh-Ritz method. Initially, the method is applied to waveguides with cross section approximated by a convex polygon. Polynomial basis functions are used. Such basis functions limit the number of modes that can be computed, since polynomials of high degree generate computer instabilities and overflow. Using the same methodology, the modal fields of superquadric waveguides are analyzed. Polynomial basis functions are replaced by trigonometric functions, however, producing an efficient numerical model capable of computing a large number of modes with a reasonable computer time. Superquadric waveguides constitute a class of uniform waveguides that include circular, elliptical, almost rectangular waveguides and a series of intermediate cross section waveguides. As a consequence they are quite useful in the construction of several devices, including transitions between waveguides of different cross sections, such as circular to elliptical and circular to rectangular. The mode matching technique is applied to the determination of the scattering matrix of discontinuities between superquadric waveguides. Several numerical applications are presented and compared to results obtained from other techniques.