The DG method has found rapid applications in such diverse areas such as aeroacoustics, electro-magnetism, gasdynamics, granular flows, magnetohydrodynamics, meteorology, modeling of shallow water, oceanography, oil recovery simulation, semiconductor device simulation, traort of contaminant in porous media, turbomachinery, turbulent flows, viscoelastic flows and weather forecasting, among many others. Discontinuous Galerkin (DG) methods belong to the Discontinuous Galerkin method is used for solving wave equation, especially for long-times simulation. In this investigation, in order to space discretization of the wave equation, the discontinuous Galerkin method was used. Error Analysis for Discontinuous Galerkin method using Fourier analysis for linear time-dependent convection equations with periodic boundary conditions was expanded and super convergence feature for a polynomial of degree k on each element was investigated. Expansion of this method for nonlinear hyperbolic differential equations is important. The superconvergence behavior of the ltr"