We develop a unified effective field theory framework for phonon dynamics in partially coherent quantum fluids, explicitly incorporating the role of a coherence parameter that encodes the fraction of the superfluid component. Starting from a microscopic description and systematically deriving the low-energy effective action, we show that coherence modifies the kinetic structure of the theory, leading to a renormalized sound speed and an emergent effective geometry governing phonon propagation. A key result of this work is the identification of a coherence dependent effective wave operator, from which the physically relevant metric is reconstructed. This resolves ambiguities associated with conformal rescalings and establishes a direct link between microscopic coherence and emergent spacetime structure. At the nonlinear level, we derive a temperature and coherence dependent effective mass for phonon excitations. Importantly, this mass does not correspond to a gap in the dispersion relation, but rather to an inertial response arising from thermodynamic gradients. We show that this effect is strongly enhanced near the superfluid transition and depends sensitively on the thermodynamic path. Order of magnitude estimates for ultracold atomic gases indicate that the effective mass can become comparable to the atomic mass in experimentally accessible regimes. This suggests that coherence-induced inertial effects may be observable through phonon dynamics in inhomogeneous or near critical systems. Our results establish a quantitative bridge between microscopic coherence, effective geometry, and emergent inertial phenomena, opening new avenues for probing collective excitations in quantum fluids.
John Mayo Moreno Faubla (Mon,) studied this question.