Full-state acoustic mode identification and shock-ripple information closure in compressible flows

In heterogeneous and shock-transmitting compressible flows, acoustic identity is a full-state dynamical property and not a property of velocity alone. Acoustic, entropy, and vortical contents may share one snapshot, while hidden pressure, density, or entropy variables alter the component selected by the linearized dynamics. We develop a full-state acoustic mode-identification framework whose parent object is the projector induced by a closed linearized compressible operator in its natural energy space. Branch projectors, admissible acoustic model-class projectors, entropy–vorticity refinements, and reduced observable projectors are treated as distinct exact objects inherited from this parent structure; the density-weighted Hodge split is retained as the kinematic fallback available from velocity data alone. The framework is verified on conservative full-state, admissible-class, and entropy–vorticity benchmarks and then applied to smooth heterogeneity and sharp shock transmission. In heterogeneous lenses, the Hodge split remains kinematically exact but separates from the full-state acoustic object when hidden thermodynamic coupling becomes active. In stationary oblique shocks, linearized Rankine–Hugoniot transmission gives machine-precision channel recovery. In finite-band shock-wavepacket holography, downstream bulk data are rank deficient, whereas adding the shock-ripple trace closes the inverse problem, reconstructs hidden upstream acoustic content, and localizes the dominant source region with zero peak-location error in both tested geometries. A robustness study shows that the augmented inverse retains full rank under severe subsampling and degrades smoothly with observation noise. The results show that shock/interface motion can act as an information-carrying observable for compressible mode conversion and inverse acoustic reconstruction.