  When immersed in a see of cold electrons, local impurities give rise to
density modulations known as Friedel oscillations. In spite of the generality
of this phenomenon, the exact shape of these modulations is usually computed
only for non-interacting electrons with a quadratic dispersion relation. In
actual materials, one needs to take into account several additional factors,
such as (i) the details of the band structure, (ii) the lifetime of
quasiparticles, (iii) in superconductors, the presence of a pairing gap.
Studying how these effects influence Friedel oscillations is a viable way to
access the properties of fermionic excitations in strongly-correlated
materials. In this work we analyze the signatures of Friedel oscillations in
STM and X-ray scattering experiments, focusing on the concrete example of
cuprates superconductors. A detailed comparison with recent experiments reveals
a rich interplay between local modulation of the chemical potential and of the
pairing gap.
