This paper reconsiders the basic manner in which the aeroacoustic equations have been derived, and proposes an alternative formulation in which we endeavor to separate ’hydrodynamic’ and ’acoustic’ terms, which we treat as synonymous with ‘radiating’ and ‘non-radiating’ mechanisms. Instead of beginning with the conservative form of the momentum and continuity equations, the analysis begins with an alternative form of them. By taking the divergence it is possible to sort the contributions to the pressure field into three separate parts, only one of which depends entirely on compressibility and vanishes identically in an incompressible flow. By considering the free space Green’s functions we argue that only this term can radiate acoustically, and the remaining terms can not. Further considerations lead to a Lighthill-like equation but with a convective d’Alembertian operator and a fully compressible source term.