  We use inelastic neutron scattering to study acoustic phonons and spin
excitations in single crystals of NaFeAs, a parent compound of iron pnictide
superconductors. NaFeAs exhibits a tetragonal-to-orthorhombic structural
transition at $T_s\approx 58$ K and a collinear antiferromagnetic (AF) order at
$T_N\approx 45$ K. While longitudinal and out-of-plane transverse acoustic
phonons behave as expected, the in-plane transverse acoustic phonons reveal
considerable softening on cooling to $T_s$, and then harden on approaching
$T_N$ before saturating below $T_N$. In addition, we find that spin-spin
correlation lengths of low-energy magnetic excitations within the FeAs layer
and along the $c$-axis increase dramatically below $T_s$, and show weak anomaly
across $T_N$. These results suggest that the electronic nematic phase present
in the paramagnetic tetragonal phase is closely associated with dynamic
spin-lattice coupling, possibly arising from the one-phonon-two-magnon
mechanism.
