  We report an exact map into one dimensional effective chains, of the
tight-binding Hamiltonian for electrons in armchair and zigzag graphene
nanoribbons with any uniaxial ripple. This mapping is used for studying the
effect of uniaxial periodic ripples, taking into account the relative
orientation changes between $\pi$ orbitals. Such effects are important for
short wavelength ripples, while for long-wave ones, the system behaves nearly
as strained graphene. The spectrum has a complex nature, akin to the Hofstadter
butterfly with a rich localization behavior. Gaps at the Fermi level and
dispersionless bands were observed, as well. The complex features of the
spectrum arise as a consequence of the quasiperiodic or periodic nature of the
effective one dimensional system. Some features of these systems are
understandable by considering weakly coupled dimers. The eigenenergies of such
dimers are highly degenerated and the net effect of the ripple can be seen as a
perturbation potential that splits the energy spectrum. Several particular
cases were analytically solved to understand such feature.
