Direct numerical simulations are performed for incompressible, turbulent channel flow over a smooth wall and different sinusoidal wall roughness configurations at a constant $Re_\tau = 720$ . Sinusoidal walls are used to study the effects of well-defined geometric features of roughness-amplitude, $a$ , and wavelength, $\lambda$ , on the flow. The flow in the near-wall region is strongly influenced by both $a$ and $\lambda$ . Establishing appropriate scaling laws will aid in understanding the effects of roughness and identifying the relevant physical mechanisms. Using inner variables and the roughness function to scale the flow quantities provides support for Townsend's hypothesis, but inner scaling is unable to capture the flow physics in the near-wall region. We provide modified scaling relations considering the dynamics of the shear layer and its interaction with the roughness. Although not a particularly surprising observation, this study provides clear evidence of the dependence of flow features on both $a$ and $\lambda$ . With these relations, we are able to collapse and/or align peaks for some flow quantities and, thus, capture the effects of surface roughness on turbulent flows even in the near-wall region. The shear-layer scaling supports the hypothesis that the physical mechanisms responsible for turbulent kinetic energy production in turbulent flows over rough walls are greatly influenced by the shear layer and its interaction with the roughness elements. Finally, a semiempirical model is developed to predict the contribution of pressure and skin friction drag on the roughness element based purely on its geometric parameters and the corresponding shear-layer velocity scale.