  Charge density waves (CDWs) are ubiquitous in under-doped cuprate
superconductors. As a modulation of the valence electron density, CDWs in
hole-doped cuprates possess both Cu-3d and O-2p orbital character owing to the
strong hybridization of these orbitals near the Fermi level. Here, we
investigate under-doped Bi$_2$Sr$_{1.4}$La$_{0.6}$CuO$_{6+\delta}$ using
resonant inelastic X-ray scattering (RIXS) and find that a short-range CDW
exists at both Cu and O sublattices in the copper-oxide (CuO2) planes with a
comparable periodicity and correlation length. Furthermore, we uncover
bond-stretching and bond-buckling phonon anomalies concomitant to the CDWs.
Comparing to slightly over-doped Bi$_2$Sr$_{1.8}$La$_{0.2}$CuO$_{6+\delta}$,
where neither CDWs nor phonon anomalies appear, we highlight that a sharp
intensity anomaly is induced in the proximity of the CDW wavevector (QCDW) for
the bond-buckling phonon, in concert with the diffused intensity enhancement of
the bond-stretching phonon at wavevectors much greater than QCDW. Our results
provide a comprehensive picture of the quasi-static CDWs, their dispersive
excitations, and associated electron-phonon anomalies, which are key for
understanding the competing electronic instabilities in cuprates.
