In this paper, we model a circular pipe with wavy inner wall, for the purpose of studying the role of surface roughness in a purely oscillating flow. Overset-grid technique is utilized for two combined flow domains, and the interpolation process within the shared zone is validated with the exact laminar flow solution for long-time oscillation. Direct numerical simulations are performed at different flow conditions, taking advantage of the overlapping capability of the spectral element method. All simulations begin with zero initial conditions, and periodic boundary conditions are applied at the two ends of the pipe with different roughness heights. The internal pipe roughness modeled by the overset meshes operates as a triggering mechanism for transition to turbulence, and the critical Reynolds number based on the Stokes thickness and the centerline velocity amplitude is determined to be 223.5 at the Stokes number of 10. The results confirm that the periodic turbulence bursts react to the presence of the roughness with different levels of turbulence intensity among the four Stokes numbers presented herein. Additionally, friction losses are calculated and compared with three cases of the existing experimental results for smooth and rough walls.