Many equations in the continuum, like the inviscid linearized shallow-water, Maxwell, or sound wave equations, can be rewritten in the form of a set of two decoupled transport equations, yielding the dispersion relation, a between relation phase speed and wavenumber. In this paper, the transport equations are discretized with the continuous Galerkin (CG) and discontinuous Galerkin (DG) methods using high-degree interpolating polynomials, and the discrete equations were analyzed through Fourier analyses. I contrast to the Fourier approaches, explicit analytical formulas have been obtained for the dispersion relations. The presence of discontinuities or gaps in the dispersion relation is characterized analytically for the CG and centered DG methods. A branch selection procedure is proposed to remove the mathematical artifacts generated by the Fourier method, leading to a single-valued dispersion relation. The presence of eventual gaps at specific wavenumbers is characterized analytically, and their specific locations are computed. Wave packets with energy at these wavenumbers will fail to propagate correctly, leading to significant numerical dispersion. An analytical ‘cleaning’ procedure results in a dispersion relation that is a single-valued function of wavenumber. Furthermore, the presence of higher frequency eigenmodes is investigated analytically and numerically, named as erratic stationary modes, which have spuriously large phase velocity errors. Upwind DG was shown to have neither spectral gaps nor an erratic stationary mode.