  Nematic order is ubiquitous in liquid crystals and is characterized by a
rotational symmetry breaking in an otherwise uniform liquid. Recently a similar
phenomenon has been observed in some electronic phases of quantum materials
related to high temperature superconductivity, particularly in the Fe-based
superconductors. While several experiments have probed nematic fluctuations,
they have been primarily restricted to the uniform nematic susceptibility, i.e.
q = 0 fluctuations. Here, we investigate the behavior of finite-momentum
nematic fluctuations by measuring transverse acoustic phonon modes with
wavelengths of up to 25 unit cells in the prototypical Fe-based compound
Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$. While the slope of the phonon dispersion gives
information about the uniform nematic susceptibility, deviations from this
linear behavior at finite but small wave-vectors are attributed to
finite-momentum nematic fluctuations. Surprisingly, these non-zero q
fluctuations lead to a softening of the phonon mode below the superconducting
transition temperature, in contrast to the behavior of the phonon velocity at q
= 0, which increases below $T_c$. Our work not only establishes a sound method
to probe long wavelength nematic fluctuations, but also sheds light on the
unique interplay between nematicity and superconductivity in Fe-based
compounds.
