Charge-density wave order is now understood to be a widespread feature of underdoped cuprate high-temperature superconductors, although its origins remain unclear. While experiments suggest that the charge-ordering wavevector is determined by Fermi-surface nesting, the relevant sections of the Fermi surface are featureless and provide no clue as to the underlying mechanism. Here, focusing on underdoped YBa$_2$Cu$_3$O$_{6+x}$, we propose a scenario that traces the charge-density wave formation to the incipient softening of a bond-buckling phonon. The momentum dependence of its coupling to the electrons in the copper-oxygen planes favourably selects the incommensurate and axial ordering wavevector found in experiments. But, it requires strong electronic correlations via their cuprate-specific renormalization of the weight and the dispersion of quasiparticles to enable a unique enhancement of the charge susceptibility near the B$_{1g}$-phonon selected wavevector. The frequency of the B$_{1g}$ phonon softens by a few percent, and a lattice instability with concomitant finite-range charge-density wave correlations will form locally, if nucleated by defects or dopant disorder. These results offer the perspective that the complex phase diagram of underdoped cuprates cannot be understood in the context of strong electronic correlations alone.
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