Modified versions of the Reynolds equation are derived with the aid of Stokes solutions for flows in channels with parallel and transverse surface roughness. The surface corrugations are of small amplitude and are represented by stationary random functions. Solutions to the modified and unmodified Reynolds equations for a wide slider bearing are presented up to the first-order terms in the slope of the film thickness. The predictions of the modified Reynolds equations, while consistent with the Stokes solutions, are qualitatively opposite to that of the unmodified Reynolds equation. Direct evidence of the inadequacy of the unmodified Reynolds equation for a two-dimensional squeeze-film bearing with parallel surface roughness is also reported.