Fresnel's Mechanical Legacy Recovered: The Drag Coefficient from Carried Compliance, and the Two Laws Bubble Acoustics Separates

Abstract

Fresnel derived the drag coefficient of moving transparent matter from a mechanical argument in 1818, and a century of interferometry confirmed it exactly as he gave it. We show that the mechanical underpinning he sought was workable in principle and wrong in one word: density, where the mechanics says compliance. The complete first-order coefficient follows from a single constitutive calculation, each component responding along its own path at the frequency it samples: $$f(\omega )=C_{\mathrm{soft}}/C_{\mathrm{total}}+(\omega /2)C_{\mathrm{soft}}^{\prime }(\omega )/C_{\mathrm{total}}.$$ The coefficient contains two laws. The first is the law Fresnel wrote: the non-dispersive part, $1-1/n^2$, equals the modified medium's share of the wave-accessible compliance, $\chi /(1+\chi )$: his excess hypothesis made exact. The second is a law of resonance, the term Lorentz derived and Zeeman confirmed. We prove the resonance law is universal mechanics: sound in a bubbly liquid obeys it in the same form and prefactor, eleven orders of magnitude below, invariantly across the analyzed two-fluid closures. The same analysis shows the share law is the carried-response term: present when the compliant channel stores and delivers in its moving frame, absent from every closure that severs response from carrier. Closure-bound cavities (geometry fixed by phase closure of the medium's waves) provide that custody by construction. One transport structure thus appears three times: light in moving glass carries both terms, the drifting bubbly liquid isolates the resonance term by slipping, and Lorentzian velocity composition reproduces the share. The direction of derivation is established: the mechanics generates what the kinematics reproduces; the composition is itself derived as the cavities' closure condition in a companion paper (arXiv:2604.27525). A documentary history traces how interpretation fused the two laws, Fresnel to Zeeman.

v3: ancillary verification script restored to its complete form (carried-response and impossibility parts); manuscript text unchanged. 20 pages, 2 tables, no figures. Cross-domain analysis connecting acoustic and optical wave propagation through compliant media. Symbolic verification script (SymPy) included as ancillary file. Companion paper: arXiv:2604.27525