We present a Fourier analysis of the first order wave equation in a periodic domain subject to a class of high-order continuous and discontinuous discretizations with either centered or upwind flux. This allows us to analytically derive the dispersion relation, group velocity and identify the emergence of gaps in the dispersion relation at specific wavenumbers. Wave packets with energy at these wavenumbers will fail to propagate correctly, and there will be significant numerical dispersion and other undesirable artifacts. Through our analysis we provide analytic formulas for the dispersion relation when approximation spaces of polynomial functions of degree n are considered. The formulas have been checked for polynomial degrees up to degree 20 for the continuous Galerkin method and up to degree 10 for the discontinuous case. We conjecture that our results hold for arbitrary polynomial degree. Such a Fourier analysis provides an alternative proof of existing results (the eventual presence of a stationary erratic mode, order of convergence, etc.). Finally, for the first time to our knowledge, the existence of gaps is characterized analytically and their specific locations are computed for both the continuous and centered discontinuous Galerkin methods. Conversely, the upwind discontinuous Galerkin method is shown to have neither spectral gaps or an erratic stationary mode.