Alternate power law velocity profile in transitional rough pipe fully turbulent flow, has been proposed, in terms of new appropriate inner rough wall variables (, ), and new parameters termed as the roughness friction Reynolds number, termed as the roughness Reynolds number and termed as roughness scale (along with normal wall coordinate where is the shift of the origin of boundary layer due to the rough wall, and ). The envelope of the power law shows that the power law constants and depend on the parameter (i.e., and ) but explicitly independent of the wall roughness parameter (roughness height in pipe of radius ). The roughness scale has been related to the roughness function of Clauser representing the velocity shift caused by wall roughness. The present results of the velocity profile, just slightly above the wall roughness level , remain valid for all types of wall roughness. The data of Nikuradse for sand-grain roughness, in transitional and fully rough pipes, has been considered, which provides good support to the predictions of an alternate power law velocity profile, based on single parameter , the roughness friction Reynolds number.