A transport geometry of acoustic analogies:exact holonomy of source re-attribution and its observable consequences
The source term of an acoustic analogy is not unique: different rearrangements of the Navier-Stokes equations attribute the same radiated sound to different apparent sources. Although this non-uniqueness has long been recognised, it has never been given a quantitative structure. We provide one by organising acoustic analogies into a fibre-like family over the space of effective media and defining transport between analogies through unique frequency-preserving, rotation-free linear space-time maps. Three exact results follow. First, the classical family of convected analogies is not closed under transport: successive uniform-flow regaugings generate effective media with anisotropic sound-speed tensors, recovering the generalised media introduced by Goldstein. Second, the discrete holonomy of analogy transport is obtained in closed form. It consists of an exact spectral dilation and a rotation that vanishes to fourth order in Mach number; transport becomes singular on a sonic horizon in analogy space. Third, in the continuum limit the boost sector is flat, while curvature is confined to anisotropy directions and is given by the commutator of sound-speed-tensor increments. The geometry does not imply any change in the physical sound field: all exact analogies yield the same far field. Its significance is operational. When an approximate source model is transported between analogies, as commonly occurs in hybrid prediction methods, the resulting far-field predictions acquire an exact, parameter-free bias consisting of a rigid spectral dilation and directivity rotation. An exact far-field law for anisotropic media extends these results to open transport paths. All identities are verified symbolically and numerically.