  A softening of phonon-dispersion has been observed experimentally in
under-doped cuprate superconductors at the charge-density wave (CDW) ordering
wave vector. Interestingly, the softening occurs below the superconducting (SC)
transition temperature T$_{c}$, in contrast to the metallic systems, where the
softening occurs usually below the CDW onset temperature T$_{\text{CDW}}$. An
understanding of the `anomalous' nature of the phonon-softening and its
connection to the pseudo-gap phase in under-doped cuprates remain open
questions. Within a perturbative approach, we show that a complex interplay
among the ubiquitous CDW, SC orders and life-time of quasi-particles associated
to thermal fluctuations, can explain the anomalous phonon-softening below
T$_{c}$. Furthermore, our formalism captures different characteristics of the
low temperature phonon-softening depending on material specificity.
