  Scanning tunneling microscopy (STM) measurements have observed modulations in
the density of states (DOS) of a number of high-T$_c$ cuprates. These
modulations have been interpreted in terms of electron-boson coupling analogous
to the dispersion "kinks" observed by angle-resolved photoemission spectroscopy
(ARPES). However, a direct a reconciliation of the energy scales and features
observed by the two probes is presently lacking. In this paper we examine the
general features of el-boson coupling in a $d-$wave superconductor using
Eliashberg theory, focusing on the structure of the modulations and the role of
self energy contributions $\lambda_z$ and $\lambda_{\phi}$. We identify the
features in the DOS that correspond to the gap-shifted bosonic mode energies
and discuss how the structure of the modulations provides information about an
underlying pairing mechanism and the pairing nature of the boson. We argue that
the scenario most consistent with the STM data is that of a low-energy boson
mode renormalizing over a second dominant pairing interaction and we identify
this low-energy mode as the out-of-phase bond buckling oxygen phonon. The
influence of inelastic damping on the phonon-modulated DOS is also examined for
the case of Bi$_2$Si$_2$CaCu$_2$O$_{8+\delta}$. Using this simplified framework
we are able to account for the observed isotope shift and anti-correlation
between the local gap and mode energies. Combined, this work provides a direct
reconciliation of the bandstructure renormalizations observed by both ARPES and
STM in terms of coupling to optical oxygen phonons.
