The Seebeck coefficient $S$ of the cuprate YBa$_{2}$Cu$_{3}$O$_{y}$ was measured in magnetic fields large enough to suppress superconductivity, at hole dopings $p = 0.11$ and $p = 0.12$, for heat currents along the $a$ and $b$ directions of the orthorhombic crystal structure. For both directions, $S/T$ decreases and becomes negative at low temperature, a signature that the Fermi surface undergoes a reconstruction due to broken translational symmetry. Above a clear threshold field, a strong new feature appears in $S_{rm b}$, for conduction along the $b$ axis only. We attribute this feature to the onset of 3D-coherent unidirectional charge-density-wave modulations seen by x-ray diffraction, also along the $b$ axis only. Because these modulations have a sharp onset temperature well below the temperature where $S/T$ starts to drop towards negative values, we infer that they are not the cause of Fermi-surface reconstruction. Instead, the reconstruction must be caused by the quasi-2D bidirectional modulations that develop at significantly higher temperature.
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