An Absolute Acoustic Readout of Plate Wave Speed for Bullion Authentication, Using a Measured Fill-Factor Correction for Struck Coins

Abstract

Acoustic "ping" testing of bullion is empirical: it compares a coin's ringing frequency with frequencies recorded for the same coin type, so it cannot assess a specimen absent from the reference library. This work replaces the comparison with an absolute measurement, computed from the measured resonance and geometry through a physical model rather than retrieved from a library. Free-plate flexural resonance is calibrated on five machined aluminium discs ($a/t=2.58$--$10.75$), recovering an empirical finite-thickness coefficient $β=0.895\pm0.047$ that reduces the worst-case departure from thin-plate theory from 8.9% to 0.75%. A fill factor, the ratio of a coin's displaced volume to its nominal cylinder, supplies the effective thickness that relief hides from calipers, extending the calibration to struck coins. The readout is the plate wave speed $c_{p}=\sqrt{E/ρ(1-ν^{2})}$, a material constant, conditioned on an assumed Poisson's ratio. On six struck 0.999 copper rounds not used in the calibration it recovers a Young's modulus of $119.1\pm6.7$ GPa, inside the accepted 110 to 128 GPa. The recovered wave speeds form distinct observed groups across the measured specimens, with the strongest independent validation provided by these copper rounds. The $(3,0)/(2,0)$ mode ratio falls below its thin-plate value in thirteen of sixteen specimens (one-sided sign test, $p=0.011$), as finite-thickness theory predicts. Weighing displaced mass to 0.1 mg rather than 0.1 g cut per-specimen volume uncertainty from 1.6% to 30% to 0.5% to 2.6% and shrank an apparent 11% design-dependent bias to a 4.7% residual on one deeply relieved silver design. The readout is now limited by tabulated elastic constants and the assumed Poisson's ratio, not by the acoustics, which are reproducible to 0.15%.