We investigate the energy-based discontinuous Galerkin (EDG) methods for solving second-order wave equations. The standard EDG formulation produces spurious oscillations near solution discontinuities and yields incorrect wave speeds when the initial data contains a discontinuity. To address these issues, we introduce an oscillation-free approach, augmented with an additional penalty term, to develop the OF-EDG method. The new formulation effectively suppresses spurious oscillations near discontinuities while preserving high-order accuracy for smooth solutions. We establish stability analysis and provide a priori error estimates for several common numerical flux choices. Through a series of numerical experiments, we demonstrate optimal convergence for smooth solutions and confirm the robustness of the OF-EDG method in maintaining oscillation-free behavior for nonsmooth solutions, both for linear wave equations and those with nonlinear source terms. Furthermore, we highlight the importance of the penalty term for ensuring convergence to the true solution when the initial data contains discontinuities.