A transport geometry of acoustic analogies:exact holonomy of source re-attribution and its observable consequences

Abstract: 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.

• The paper formalizes acoustic-analogy re-attribution as frequency-preserving transport and proves that classical convected analogies are not compositionally closed, requiring anisotropic effective media for closure.
• The paper derives exact holonomy as a frequency dilation and spatial rotation, including a 36% spectral displacement for two equal collinear re-gaugings at Mach 0.45 and a sonic horizon at Mach 0.618 for equal steps.
• The paper shows that off-shell hybrid prediction pipelines obey the exact law I₂(ŝ,ω)=I₁(Rŝ,ρω), while continuum boost transport is frequency-flat and curvature localizes in noncommuting anisotropy changes.

Overview

This paper, by Sparsh Sharma (DLR Braunschweig), gives a quantitative structure to the long-recognised non-uniqueness of acoustic-analogy source attribution. Acoustic analogies—exact rearrangements of the Navier–Stokes equations into $L\varphi = s$ with everything not in the wave operator declared "source"—are organised into a fibre-like family over constant-coefficient effective media. The operation of passing between analogies ("re-gauging") is formalised as transport by the unique frequency-preserving, rotation-free linear space–time maps reducing each convected wave operator to the d'Alembertian. Three exact results emerge: the classical convected analogies are not closed under this transport; the discrete holonomy of closed loops consists of an exact frequency dilation $\rho$ and a rotation that vanishes to fourth order in Mach number; and in the continuum limit all curvature localises in the anisotropy directions of the effective medium.

Exact equivalence and the kinematic cocycle

The paper begins by making precise why the seventy-year debate over "true sources" could not be resolved on its own terms. Under the definition of an exact analogy, the far-field trace of the wave variable equals the physical far-field pressure of the solution itself, so all exact analogies are on-shell far-field equivalent—the classification of exact analogies by their far fields is empty. Any meaningful comparison must therefore be made off shell, where modelled sources are inserted into solution operators.

The one nontrivial fact at the exact level is the transformation law between analogies sharing a dependent variable: $s_2 - s_1 = (L_2 - L_1)\varphi$, which holds identically on solutions. For the Lighthill-to-convected-Lighthill re-gauging, a stronger purely kinematic identity holds off shell: for arbitrary smooth fields satisfying only mass conservation,

$$\partial_i\partial_j\bigl(T^0_{ij}-T^U_{ij}\bigr) + (\Box_U - \Box_0)\rho = 2(U\cdot\nabla)\,\mathcal{C},$$

where $\mathcal{C}$ is the continuity residual. Re-attribution requires no momentum equation or constitutive relation—it is bookkeeping, not dynamics—and the residual $2(U\cdot\nabla)\mathcal{C}$ measures exactly how much pipeline disagreement is attributable to model inconsistency with mass conservation. This identity makes rigorous the empirical observations of Samanta et al., who found identical far fields from several analogies driven by identical data but analogy-dependent degradation once input errors were introduced.

Frequency-respecting maps and canonical reduction

Because prediction is performed frequency by frequency, transport maps must preserve the time-harmonic decomposition. Such maps are characterised exactly as block upper-triangular matrices with spatial rows containing no time component; they form a group and rescale frequency $\omega \mapsto \omega/p$. The Galilean change to the medium frame fails this condition; the admissible reduction is instead the Prandtl–Glauert–Lorentz-type map, whose space–time factorisation is structural:

$w(v) = \ell(-v)\,g(v),$

a Galilean change of frame completed by a Lorentz boost at signal speed $c$. This factorisation satisfies $w(v)\,Q_c(v)\,w(v) = *$ exactly, and its time-harmonic restriction reproduces the classical fixed-frequency reduction with phase $\rho$0 forced uniquely by elimination of the first-derivative term. The conformal stabiliser of the d'Alembertian within the block group is exactly $\rho$1, fixing the residual freedom as a frequency scaling plus rigid rotation.

Non-closure and the forced generalised media

The central structural result is that the family of classical convected operators $\rho$2 is not closed under composed transport. Two perpendicular subsonic re-gauging steps produce an endpoint cometric

$\rho$3

with anisotropic sound-speed tensor $\rho$4 satisfying $\rho$5 exactly and diagonal subsonicity tensor $\rho$6. Collinear steps are equally non-classical after normalisation. Consequently, the anisotropic effective media of Goldstein's generalised acoustic analogy are forced as a closure requirement rather than adopted as a modelling convenience—the minimal transport-closed enlargement of the classical family. Each admissible medium $\rho$7 possesses a unique rotation-free canonical reduction $\rho$8 with $\rho$9, $s_2 - s_1 = (L_2 - L_1)\varphi$0, $s_2 - s_1 = (L_2 - L_1)\varphi$1, recovering the Prandtl–Glauert–Lorentz map in the isotropic case.

The collinear composition law is notable: the endpoint velocity obeys neither Galilean nor Einstein addition, agreeing with the latter only for perpendicular steps. It is a hybrid, Lorentzian transversely and Galilean-contaminated longitudinally ($s_2 - s_1 = (L_2 - L_1)\varphi$2).

Exact holonomy and the sonic horizon

For any composite transport $s_2 - s_1 = (L_2 - L_1)\varphi$3 with admissible endpoint, the discrepancy between the composite and direct canonical reduction of the endpoint—the holonomy—is exactly

$s_2 - s_1 = (L_2 - L_1)\varphi$4

a rigid frequency dilation composed with a rigid spatial rotation. For two boost steps the dilation is given in closed form:

$s_2 - s_1 = (L_2 - L_1)\varphi$5

depending only on $s_2 - s_1 = (L_2 - L_1)\varphi$6 and the invariant $s_2 - s_1 = (L_2 - L_1)\varphi$7. Three consequences follow immediately. First, $s_2 - s_1 = (L_2 - L_1)\varphi$8 exactly for perpendicular steps, which coincide with relativistic velocity composition. Second, because $s_2 - s_1 = (L_2 - L_1)\varphi$9 contains $$\partial_i\partial_j\bigl(T^0_{ij}-T^U_{ij}\bigr) + (\Box_U - \Box_0)\rho = 2(U\cdot\nabla)\,\mathcal{C},$$0 but not $$\partial_i\partial_j\bigl(T^0_{ij}-T^U_{ij}\bigr) + (\Box_U - \Box_0)\rho = 2(U\cdot\nabla)\,\mathcal{C},$$1, exchanging step order changes the holonomy at fourth order: transport is irreducibly path-ordered. Third, the boost-pair rotation angle is $$\partial_i\partial_j\bigl(T^0_{ij}-T^U_{ij}\bigr) + (\Box_U - \Box_0)\rho = 2(U\cdot\nabla)\,\mathcal{C},$$2—fourth order, parametrically smaller than the second-order Thomas–Wigner rotation of pure Lorentz boosts, vanishing identically for collinear and perpendicular steps.

Transport degenerates on an exact sonic horizon in analogy space: admissibility holds if and only if $$\partial_i\partial_j\bigl(T^0_{ij}-T^U_{ij}\bigr) + (\Box_U - \Box_0)\rho = 2(U\cdot\nabla)\,\mathcal{C},$$3. At equality $$\partial_i\partial_j\bigl(T^0_{ij}-T^U_{ij}\bigr) + (\Box_U - \Box_0)\rho = 2(U\cdot\nabla)\,\mathcal{C},$$4 and $$\partial_i\partial_j\bigl(T^0_{ij}-T^U_{ij}\bigr) + (\Box_U - \Box_0)\rho = 2(U\cdot\nabla)\,\mathcal{C},$$5; in the collinear case the entire subsonicity tensor degenerates simultaneously through the shared factor $$\partial_i\partial_j\bigl(T^0_{ij}-T^U_{ij}\bigr) + (\Box_U - \Box_0)\rho = 2(U\cdot\nabla)\,\mathcal{C},$$6. In Mach numbers the collinear boundary is $$\partial_i\partial_j\bigl(T^0_{ij}-T^U_{ij}\bigr) + (\Box_U - \Box_0)\rho = 2(U\cdot\nabla)\,\mathcal{C},$$7, giving for equal steps the inverse golden ratio $$\partial_i\partial_j\bigl(T^0_{ij}-T^U_{ij}\bigr) + (\Box_U - \Box_0)\rho = 2(U\cdot\nabla)\,\mathcal{C},$$8. Physically, two individually subsonic re-gaugings compound to an effectively supersonic one; the paper is careful to state that this is a degeneration of the uniform-coefficient transport, not a claim of physical shock formation.

Continuum limit: flat boosts, curved anisotropy

In the continuum limit the frequency connection form is exact: $$\partial_i\partial_j\bigl(T^0_{ij}-T^U_{ij}\bigr) + (\Box_U - \Box_0)\rho = 2(U\cdot\nabla)\,\mathcal{C},$$9, so continuum transport multiplies frequencies by $\mathcal{C}$0 independently of path, and every smooth loop has trivial frequency holonomy. The discrete two-step holonomy is precisely the finite-step composition defect. Since the per-step rotation is quadratic in the increment, the boost sector is continuum-flat.

The Lie-algebraic mechanism is explicit: the boost generator $\mathcal{C}$1 is strictly upper-triangular because the Galilean factor cancels exactly the spatial–time column of the Lorentz generator—the entries responsible for Thomas–Wigner rotation. Curvature localises entirely in the anisotropy directions, where it equals the commutator of sound-speed-tensor increments divided by $\mathcal{C}$2: nonzero exactly when medium deformations fail to commute (e.g. a normal stretch and a 45° shear). A discrete check confirms the stretch–shear triangle holonomy angle $\mathcal{C}$3 numerically to convergence. The connection also carries torsion on mixed boost–anisotropy planes, whose physical meaning the paper flags as open.

Observable consequences for hybrid prediction pipelines

The geometry becomes falsifiable when it acts on statistical source models. The paper is emphatic about what is and is not observable: on shell nothing changes—no transport alters any measurable quantity of radiated sound, and reading $\mathcal{C}$4 as a physical frequency shift would be incorrect. What the holonomy quantifies is the discrepancy incurred off shell when one and the same modelled source spectrum is transported through different analogies, as routinely happens in hybrid prediction. For stationary random sources in three dimensions, amplitude weights cancel identically and the cross-spectral density transports by rigid dilation and rotation of its arguments. The resulting two-pipeline law is exact and parameter-free:

$\mathcal{C}$5

Two groups modelling the same turbulence with the same source model but different analogies will predict peak frequencies differing by exactly $\mathcal{C}$6 and directivity patterns rotated by exactly $\mathcal{C}$7, with no freedom to reconcile them short of recalibration. The channels separate exactly between sectors: anisotropy loops produce pure rigid rotations ($\mathcal{C}$8), boost loops produce pure spectral dilations ($\mathcal{C}$9), matching the curvature localisation.

The magnitudes are substantial. Two equal collinear steps at $2(U\cdot\nabla)\mathcal{C}$0 give $2(U\cdot\nabla)\mathcal{C}$1—a 36% (≈0.44-octave) systematic displacement of every spectral feature—and $2(U\cdot\nabla)\mathcal{C}$2 diverges at the horizon. Numerical verification against the full transport pipeline agrees with the closed forms to machine precision (e.g. $2(U\cdot\nabla)\mathcal{C}$3 computed and predicted identically for $2(U\cdot\nabla)\mathcal{C}$4, $2(U\cdot\nabla)\mathcal{C}$5; rotation residual $2(U\cdot\nabla)\mathcal{C}$6).

Removing the restriction to quiescent endpoints required deriving, in closed form, the far field of an admissible generalised medium: the radiated field samples the source spectrum on the outgoing sonic ellipsoid $2(U\cdot\nabla)\mathcal{C}$7 along group rays, at the unique wavevector whose group velocity points at the observer. Validated end-to-end against a direct limiting-absorption FFT solution (0.35% amplitude, 0.033 rad phase agreement), this lemma extends the pipeline law to open paths via a conjugated direction map with the solid-angle Jacobian required by power conservation along ray tubes.

Limitations and open problems

The paper states its restrictions plainly. All exact computations concern spatially uniform effective media; genuinely sheared base flows would replace the matrix group by Fourier integral operators, and the uniform theory is described as the principal-symbol shadow of that theory. The clean amplitude cancellation in the spectral transport is specific to $2(U\cdot\nabla)\mathcal{C}$8; planar problems carry a residual $2(U\cdot\nabla)\mathcal{C}$9 weight. Tensor (Lighthill-type) sources acquire additional $\omega \mapsto \omega/p$0 index action, deferred here. The mapping from modelled Reynolds-stress anisotropy to effective-medium anisotropy remains an open modelling link, so rotation magnitudes are parametric where dilation magnitudes are not. Five problems are listed explicitly: proof of horizon evasion by refined paths (conjectured but unproven); extension to tensor sources; physical interpretation of the mixed-plane torsion; the variable-coefficient theory; and the identification of microlocal invariants (H-measures) of statistical source models under re-gauging.

Conclusion

The paper converts the qualitative observation that acoustic-analogy sources are non-unique into an exact geometric structure: a fibre-like family of analogies over effective media, canonical frequency-preserving transport, computed holonomy, and a sharp division between flat directions (where the choice of analogy is gauge in the strict sense) and curved directions (where no path-independent identification of source models exists). Its practical content is a parameter-free law for inter-method bias in hybrid noise prediction—a rigid spectral dilation reaching 36% for two Mach-0.45 re-gauging steps—which is verifiable, systematic, and independent of any change in the physics of the radiated sound.