  In order to identify the mechanism responsible for the formation of
charge-density waves (CDW) in cuprate superconductors, it is important to
understand which aspects of the CDW's microscopic structure are generic and
which are material-dependent. Here, we show that, at the local scale probed by
NMR, long-range CDW order in YBa2Cu3Oy is unidirectional with a commensurate
period of three unit cells (lambda = 3b), implying that the incommensurability
found in X-ray scattering is ensured by phase slips (discommensurations).
Furthermore, NMR spectra reveal a predominant oxygen character of the CDW with
an out-of-phase relationship between certain lattice sites but no specific
signature of a secondary CDW with lambda = 6b associated with a putative
pair-density wave. These results shed light on universal aspects of the cuprate
CDW. In particular, its spatial profile appears to generically result from the
interplay between an incommensurate tendency at long length scales, possibly
related to properties of the Fermi surface, and local commensuration effects,
due to electron-electron interactions or lock-in to the lattice.
