Dispersion and dissipation of unphysical modes in high-order discontinuous Galerkin (DG) methods are investigated through the Fourier analysis for the one-dimensional scalar advection equation. A combined mode consisting of the physical and unphysical mode is examined. The dispersion and dissipation of the unphysical mode are totally different with those of the physical mode, and the switching between the physical and unphysical modes exists in the combined mode. The unphysical mode is dominant in the combined mode when an input wave number is high. The analysis of the dispersion and dissipation of the combined mode then clarifies that their drastically fluctuations in time when the switching occurs. Through time averaging the dispersion and dissipation of the combined mode, a sudden deviation from exact solutions and the small dissipation for high wave numbers are found. Finally, a simple advection of a sine wave is computed to validate the present Fourier analysis.