  Although charge-density wave (CDW) correlations exist in several families of
cuprate supercon-ductors, they exhibit substantial variation in CDW wavevector
and correlation length, indicating a key role for CDW-lattice interactions. We
investigated this interaction in La$_{1.875}$Ba$_{0.125}$CuO$_4$ using single
crystal x-ray diffraction to collect a large number of CDW peak intensities,
and determined the Cu and La/Ba atomic distortions induced by the formation of
CDW order. Within the CuO$_2$ planes, the distortions involve a periodic
modulation of the Cu-Cu spacing along the direction of the ordering wave
vector. The charge ordering within the copper-oxygen layer induces an
out-of-plane breathing modulation of the surrounding lanthanum layers, which
leads to a related distortion on the adjacent copper-oxygen layer. Our result
implies that the CDW-related structural distortions do not remain confined to a
single layer but rather propagate an appreciable distance through the crystal.
This leads to overlapping structural modulations, in which CuO$_2$ planes
exhibit distortions arising from the orthogonal CDWs in adjacent layers as well
as distortions from the CDW within the layer itself. We attribute this striking
effect to the weak c-axis charge screening in cuprates and suggest this effect
could help couple the CDW between adjacent planes in the crystal.
