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

Sylvia Dong, Takyiu Liu

arXiv:2610.29194v1eess.ASphysics.class-ph

TL;DR

Acoustic ping tests compare a coin’s frequency with a coin-specific library, leaving no absolute readout for specimens outside it. This work uses a fill-factor correction to estimate plate wave speed from resonance and geometry; on six struck copper rounds, it recovers a modulus within the accepted range, while the classification scope remains limited by the absence of a blind trial.

  • Problem

    Acoustic ping testing compares specimen frequencies with coin-specific reference libraries, while a bulk elastic measurement on an object of unknown effective geometry remains missing.

    Source: §I · p. 2

  • Method

    A displacement-based fill factor supplies effective thickness for the physical resonance model, which computes plate wave speed from measured resonance and geometry.

    Source: §VI · p. 14

  • Results

    119.1 ± 6.7 GPa was the recovered Young’s modulus for six struck copper rounds, within the accepted 110–128 GPa range.

    Source: §VI · p. 14

  • Takeaways & Limitations

    The method converts acoustic comparison into an absolute wave-speed readout, with distinct observed groups across measured specimens and strongest independent validation from the copper rounds.

    Source: §VI · p. 14

    Without specimens relabelled by a third party and classified without identity knowledge, the work demonstrates separation of known specimens, not a validated classifier.

    Source: F. Limitations · p. 14

Abstract

from arXiv · show

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 $\beta=0.895\pm0.047$ 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 $c_{p}=\sqrt{E/\rho(1-\nu^{2})}$, 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%.

I. INTRODUCTION

Bullion fakes can match published physical specifications, while existing tests have specific limits and acoustic ping methods compare frequencies against coin-specific libraries. This work aims to recover a bulk elastic property from resonance despite a struck coin’s unknown effective geometry.

I. INTRODUCTION

  • Bullion fakes can match a coin’s published diameter, thickness, mass, and appearance, while density-matched tungsten can pass specific-gravity testing.X-ray fluorescence is limited to the surface, whereas ultrasonic pulse-velocity testing addresses bulk elasticity but requires a contact transducer, couplant, and dedicated instrument.Source: §I · p. 1
  • A prior acoustic system separated four possible counterfeits from thirty authentic coins using a frequency-band threshold, while consumer apps are optimized for supported coin lists.Privately minted rounds with varying specifications are reported as less reliably tested by the most developed app.Source: §I · p. 1
  • Acoustic ping methods measure a specimen-specific frequency, so changing coin geometry changes the frequency and requires a reference library for each coin type.The paper identifies the missing capability as a bulk elastic measurement on an object of unknown effective geometry without contact instrumentation.Source: §I · pp. 1–2

II. THEORY

  • Free-plate flexural resonance relates frequency to plate geometry and material properties, with a mode coefficient depending on nodal pattern and Poisson’s ratio.The fundamental free-disc mode is (2, 0); the paper recovers the extensional plate wave speed as the material factor, not the dispersive flexural-wave phase velocity.Source: §II · p. 2

A. Two obstacles

Two obstacles prevent applying thin-plate acoustic methods directly to struck coins: finite-thickness effects and relief that makes the vibrating plate’s effective thickness unmeasurable by calipers.

1) Thin-plate theory fails at coin proportions:

  • Kirchhoff thin-plate theory neglects transverse shear and rotary inertia, whose importance increases as thickness rises relative to radius.A one-ounce silver round has a/t ≈6, within the region where uncorrected theory carries meaningful error.Source: 1) Thin-plate theory fails at coin… · p. 2
  • The fitted coefficient β absorbs transverse shear and rotary inertia into a single empirical finite-thickness correction motivated by Mindlin theory.It is a one-parameter rational approximation, not a shear-correction factor in Mindlin theory’s own sense.Source: 1) Thin-plate theory fails at coin… · p. 2

2) A struck coin is not a uniform disc:

  • Relief removes material from the nominal cylinder, making caliper thickness overstate the vibrating plate’s thickness by a design-dependent amount.The fill factor uses displaced volume to supply a measurable, volume-equivalent effective thickness; it is an average rather than a dynamically equivalent thickness because local flexural rigidity varies as h^3.Source: 2) A struck coin is not a uniform disc · p. 2

B. The estimator and its sensitivities

The estimator inverts the corrected plate-resonance model using measured quantities, while the choice of Poisson’s ratio conditions the resulting wave-speed estimate.

B. The estimator and its sensitivities

  • For struck coins, the readout depends on f1, radius, displaced volume, λ2, and β; caliper thickness cancels and specimen mass does not enter.Mass instead enters separately through Archimedes density for the decision rule.Source: B. The estimator and its sensitivities · p. 2
  • On struck coins, displaced volume propagates with sensitivity −0.96 to −1.00, while radius sensitivity is +3.9 to +4.0 because radius enters both the plate geometry and volume-derived thickness.The estimator’s logarithmic sensitivity to β is q/(1 + q), and λ2 has sensitivity −1.Source: B. The estimator and its sensitivities · p. 2

C. Choice of λ2

  • Because λ2 depends on Poisson’s ratio and a specimen’s ν is unknown, the reported cp is a ν-conditioned estimate, with systematic uncertainty bounded at ±3% across ν = 0.28–0.37.The paper’s use of “absolute” means computed from measured resonance and geometry through a physical model, not an assumption-free determination of intrinsic elastic constants.Source: C. Choice of λ2 · p. 3
  • Using λ2 = 5.253 across materials introduces systematic error of up to 2.4%, largest on silver.The table of free-plate coefficients is computed from the characteristic equation; the solver reproduces Leissa’s tabulated values within 0.04% and 0.09% at the two cited Poisson’s ratios.Source: C. Choice of λ2 · p. 3, Table I · p. 3

D. Reference values

Reference elastic constants span ranges, so the paper evaluates recovered moduli against accepted ranges rather than a single selected wave-speed value. The specimen set includes known-composition references and bullion-like rounds of unverified composition.

D. Reference values

  • Tabulated Young’s moduli span 110–128 GPa for copper and 76–83 GPa for silver, so implied reference cp values inherit uncertainty before measurement.Agreement closer than the tabulated spread cannot establish accuracy by comparison with one chosen value inside the range.Source: D. Reference values · p. 3
  • The method instead recovers the modulus implied by measured cp and specimen density, then tests whether it falls within the accepted range for the claimed material.This avoids dependence on which end of a tabulated range is quoted and makes the test falsifiable.Source: D. Reference values · p. 3

A. Specimens

  • Five aluminium calibration discs span a/t from 2.58 to 10.75, with two-dimensional variation in diameter and thickness.The supplier’s alloy designation and temper were unrecorded, but both tested Poisson’s-ratio conventions yield recovered moduli within the stated aluminium range; the alloy question is immaterial at this work’s precision.Source: A. Specimens · p. 3
  • The reference set includes copper rounds, silver specimens, and a tungsten disc; five additional bullion-like rounds have unverified composition and provenance.Their measured properties are described only as consistent with candidate materials, not as material identifications.Source: A. Specimens · pp. 3–4

B. Metrology

The metrology uses liquid displacement to estimate specimen volume and validates the rig against machined discs. The uncertainty budget shows that finer displaced-mass measurements reduce volume uncertainty, while unmeasured systematic and model errors remain outside the reported per-specimen uncertainty.

B. Metrology

  • Displaced volume was measured by suspending each specimen in distilled water and averaging three balance readings; the finer-resolution readings were used when available.The displaced-mass uncertainty combines the standard deviation of three readings with a 0.5% systematic floor; specimens not remeasured retain one 0.1 g division of uncertainty.Source: B. Metrology · p. 4
  • Machined-disc checks show small volume discrepancies, while the fine-silver rounds average 10 559 ± 21 kg/m3, 0.65% above the handbook value.The 0.5% systematic floor is a judgement based on one machined disc and one material, not an independent calibration.Source: B. Metrology · p. 4
  • If the fine-silver excess reflects a common displacement bias, correcting it would lower recovered moduli but leave density decisions and group ordering unchanged.The correction is not applied because the offset has not been independently calibrated; the authors report that copper would remain within its accepted modulus range, while fine silver and C1–C4 would fall below theirs.Source: B. Metrology · p. 4

C. Uncertainty budget

  • At 0.1 mg resolution, displaced-volume uncertainty is 0.5% to 2.6%, versus a term one to two orders of magnitude larger than frequency uncertainty at 0.1 g.At finer resolution, volume remains the largest instrumental term for reference specimens, but is comparable to or smaller than reference-value and λ2 uncertainty when ν is unknown.Source: C. Uncertainty budget · p. 4
  • The reported measurement-derived uncertainty omits unknown-material Poisson’s-ratio effects, non-uniform-relief model error, displacement bias beyond the floor, and β-transfer error.Caliper thickness and specimen mass do not enter the acoustic readout; mass enters separately through Archimedes density.Source: C. Uncertainty budget · pp. 4–5

D. Acoustic apparatus

The apparatus centers and strikes specimens for the (2, 0) mode, while the analysis pipeline detects, screens, and averages taps; independent sessions agree on the calibration constant to 0.07%.

D. Acoustic apparatus

  • The specimens rested on a centered fixture at the (2, 0) mode’s node and were struck at the rim, an antinode.A foam-lined enclosure reduced reflections while leaving the front open for striker access.Source: D. Acoustic apparatus · p. 5

E. Analysis pipeline

  • 0.35 s after onset, the pipeline takes a Hann-windowed segment and zero-pads it to 262,144 samples for spectral interpolation.The segment begins 6 ms after onset; zero-padding does not improve the 2.9 Hz true spectral resolution.Source: E. Analysis pipeline · pp. 5–6
  • 0.008% to 0.084% strike-to-strike scatter, rather than the 0.366 Hz bins, is the operative bound on refined peak-position precision.Peak positions are refined by parabolic interpolation on the log-magnitude spectrum.Source: E. Analysis pipeline · p. 6
  • f1 is the lowest peak clearing 15 dB local prominence and 3% of the strongest peak, above a 1,000 Hz floor.Taps are rejected if f1 has less than 25 dB prominence or differs by more than 2% from the recording median; 89.6% of 2,163 taps were accepted.Source: E. Analysis pipeline · p. 6
  • A sufficiently damped specimen can be acoustically invisible regardless of strike strength because the prominence statistic imposes a frequency-dependent Q floor.The empirical tests detected every injected frequency by Q ≈220, while no amplitude up to half scale was detected at Q ≤100 for 16–20 kHz or Q ≤150 for 22–24 kHz.Source: E. Analysis pipeline · p. 6

A. Repeatability

  • 0.07% agreement in the fitted calibration constant was obtained from independent mountings on different days using the same estimator and parameters.The four remounted discs shifted by −0.026% to −0.222%; strike-to-strike scatter had a median of 0.050% over 96 coin takes.Source: A. Repeatability · p. 6

B. Calibration on aluminium

A fitted finite-thickness correction removes the structured aluminium calibration residuals, and the resulting readout is checked against aluminium and struck-metal specimens. The copper specimens provide the strongest independent validation, while the other observed material groupings have qualifications.

B. Calibration on aluminium

  • β = 0.895±0.047 reduces the calibration’s worst-case residual to 0.75%, from structured uncorrected residuals spanning 12.6 percentage points.The coefficient is empirical for the five discs and their aspect-ratio range, not a universal or independently validated physical constant.Source: B. Calibration on aluminium · p. 7, Figure 3 · p. 7
  • The fitted calibration is stable across analysis revisions: β changes from 0.895 to 0.897 and C by 0.03%, with worst-case residuals of 0.75% and 0.83%.Disc 4, the extreme-aspect-ratio point anchoring the thin end, has the largest input difference between files at 0.36%.Source: B. Calibration on aluminium · p. 7
  • 5,412±25 m/s is the aluminium calibration’s recovered plate wave speed, a −0.44% deviation from the 5,436 m/s handbook value.The per-disc mean implies E = 69.56 GPa against 70 GPa; the assumed Poisson’s ratio contributes a stated systematic.Source: B. Calibration on aluminium · p. 7

C. Material bands

  • Applying effective thickness teff = φt with per-material λ2 yields fill factors from 0.70 to 0.99 across the specimens.The machined sterling disc reads 0.99, while the most deeply relieved rounds read 0.70.Source: C. Material bands · p. 7
  • The independent copper band mean lands within 0.9% of handbook, with individual specimens scattering by 2.6% about it.The copper point estimates are separated from the silver readings, but the observed groups do not establish uncertainty-resolved composition classes; the silver groups are not resolved from each other.Source: Figure 4 · p. 7, C. Material bands · p. 7

RECOVERED MODULUS FALLS INSIDE THE ACCEPTED RANGE, NOT

The six struck copper rounds recover a Young’s modulus inside the accepted range, providing an independent check of the fill-factor correction. Higher-mode measurements also support the predicted finite-thickness direction, though the ratio does not resolve material classes.

RECOVERED MODULUS FALLS INSIDE THE ACCEPTED RANGE, NOT

  • 119.1±6.7 GPa falls inside the accepted 110 GPa to 128 GPa range for the six copper rounds, which independently test the fill-factor correction.Aluminium supplied the fitted coefficient, so copper is the independent test.Source: RECOVERED MODULUS FALLS INSIDE THE… · p. 8
  • At 0.1 g resolution, three deeply relieved specimens were 8% to 13% below their accepted ranges; finer-balance measurements removed most of that pattern, leaving only the brass-like round clearly low at −4.9% against generic brass.Fine silver and steel-like rounds were 1.2% and 0.3% below their ranges, respectively, each within its standard error.Source: RECOVERED MODULUS FALLS INSIDE THE… · p. 8

D. Complete specimen results

  • The silver specimens’ recovered cp was 2 790 m/s to 3 040 m/s, about 60% below the 4 810 m/s predicted from handbook constants for pure tungsten; no tungsten acoustic reading was obtained.Source: RECOVERED MODULUS FALLS INSIDE THE… · p. 8

E. Higher modes

  • Higher-frequency peaks follow free-plate modal ratios more often than integer multiples: 83.7% of 1970 peaks are within 2% of a free-plate ratio, versus 10.4% of integer multiples.The mean distance to the nearest free-plate mode is 0.044, versus 0.268 to the nearest integer.Source: E. Higher modes · p. 9
  • The dominant overtone is the (3, 0) mode at 2.23–2.35 f1; the 3.7–4.1 f1 group is reported jointly because its modes are separated by only 2.6%, against about 1% specimen scatter.The (0, 1) mode appears in only 9 of 1207 accepted taps; its rarity is attributed most plausibly, but not demonstrably, to centre-pinching by the mount.Source: E. Higher modes · p. 9
  • The sterling disc’s (3, 0)/(2, 0) ratio identifies its 2 087 Hz peak as an overtone, confirming the lower fundamental without relying on judgement.A lowered-floor rerun directly recovered a split doublet at 886.4 and 896.9 Hz.Source: E. Higher modes · p. 9
  • Thirteen of sixteen specimens lie below their material’s Kirchhoff (3, 0)/(2, 0) ratio, as finite-thickness theory predicts (one-sided sign test, p = 0.011).Excluding the two aluminium calibration discs, eleven of fourteen remain below (p = 0.029); the predicted scaling is less secure and rests mainly on Al5.Source: E. Higher modes · p. 9, Figure 7 · p. 9
  • The higher-mode ratio is not a material discriminant: theory separates copper from silver by 0.0054, versus an observed 0.031 difference dominated by geometry and relief.The theoretical ratio spans only 0.6% across the studied materials.Source: E. Higher modes · p. 10

F. A frequency-floor artifact

A 1 000 Hz peak-search floor can misassign a specimen’s fundamental when it rings below that threshold, a problem diagnosed for one sterling disc. In application, the density rule separates the measured silver specimens from copper and the unverified rounds, while the acoustic rejection of tungsten remains predicted rather than demonstrated.

F. A frequency-floor artifact

  • The 1 000 Hz peak-search floor misassigns the fundamental of the machined sterling disc, whose reported 2 087 Hz peak is its (3, 0) mode.Lower-floor results are used for that specimen throughout; no other specimen has a fundamental within 2 kHz of the floor.Source: F. A frequency-floor artifact · p. 10

G. Application

  • The decision rule admits all eight silver specimens and rejects all six copper rounds with displacement measurements and all five unverified rounds as silver.The rule is silver if non-ferromagnetic and Archimedes density exceeds 9 000 kg/m3; the magnet filters four unverified rounds and density rejects the fifth.Source: G. Application · p. 11
  • The acoustic readout gives an elasticity contrast on every call, but both it and density use the same displaced volume, so their errors are not statistically independent.The unverified rounds’ wave speeds are consistent with steel for the four ferromagnetic examples and brass for the non-ferromagnetic example.Source: G. Application · p. 11
  • The density rule admits the tungsten disc, while the predicted acoustic rejection remains untested because neither tungsten specimen yielded a reproducibly detectable resonance.The predicted rejection lies above the highest copper reading, but the non-detection does not show that the specimen lacks an elastic resonance; the sensitivity bound is conditional on frequency, damping, and excitation.Source: G. Application · p. 11
  • At matched 39.1 mm to 39.7 mm diameter, the four groups separate by fundamental frequency, but differing thickness and relief prevent a geometry-independent classification claim.Their volume-equivalent thicknesses span 2.44 mm to 3.03 mm, enough by itself to move f1 by about 24%.Source: G. Application · p. 11, Figure 8 · p. 12

H. What the unverified rounds are consistent with

The unverified rounds’ density and wave speed are consistent with plain-carbon steel or a brass-like copper–zinc alloy, but the measurements do not establish composition. On one deeply relieved silver design, finer volume measurement reduces a large apparent bias to a smaller residual that remains scoped to that family.

H. What the unverified rounds are consistent with

  • The four ferromagnetic rounds are consistent with plain-carbon steel, and the non-ferromagnetic fifth with a brass-like copper–zinc alloy, but the measurements establish no alloy grade or elemental composition.The steel-like rounds’ 2% to 5% low wave-speed readings are comparable to their uncertainty and cannot distinguish a compositional difference from a design effect.Source: H. What the unverified rounds are… · p. 12

A. A residual design dependence

  • At 0.1 mg resolution, the deeply relieved silver family averages −2.0% against 2.7% for the other specimens; the 4.7% difference relies on two coarser-resolution measurements.Among the six specimens measured at finer resolution, the difference is 3.6% and not significant (p = 0.19).Source: A. A residual design dependence · p. 12, Figure 9 · p. 12
  • A volume-equivalent uniform plate need not match a relieved specimen’s resonance, because material placement relative to modal curvature affects the Rayleigh quotient’s numerator.The sign of the discrepancy depends on relief pattern, not relief depth alone.Source: A. A residual design dependence · pp. 12–13
  • Copper rounds show no comparable bias, but all five deep-relief silver specimens share one design, so design cannot be separated from die.The finding is scoped to that family, not relief depth in general; relief depth was not quantified.Source: A. A residual design dependence · p. 13
  • On the one deeply relieved design measured, the readout averages 2.0% low and is at most 3.7% low; machined and shallow-relief pieces are unbiased within a few per cent per specimen.The six copper rounds are unbiased to 0.9% in the mean.Source: A. A residual design dependence · p. 13

B. The error budget points away from the acoustics

Measurement error is dominated by displaced volume rather than acoustic frequency, while manufacturing variation also limits how narrow an authentic band can be. The readout therefore remains a few-per-cent estimate for unknown coins, conditioned on Poisson’s ratio and tabulated elastic constants.

B. The error budget points away from the acoustics

  • At 0.1 mg rather than 0.1 g resolution, volume uncertainty falls from 1.6% to 30% per specimen to 0.5% to 2.6%, shrinking the apparent design bias from 11% to under 4%.Frequency is reproducible to 0.15% between sessions; the binding terms are now tabulated elastic-constant spread and the assumed Poisson’s ratio.Source: B. The error budget points away from… · p. 13
  • Nominally identical one-ounce copper rounds show a 3.6% frequency spread, so manufacturing variation contributes to the width of any authentic band alongside measurement.The absolute readout’s advantage is a predicted rather than learned band location, not a narrower band.Source: B. The error budget points away from… · p. 13

C. Damping is a design signature, not a material one

  • Within fine silver, Q varies from 1456 to 7013 and damping varies fivefold with design, while cp varies by a few per cent; Q is therefore excluded from the decision rule.Splitting of the degenerate (2, 0) mode measures axial asymmetry rather than composition.Source: C. Damping is a design signature, not… · p. 13

D. Why no threshold on cp

  • The readout is not given an acceptance threshold because design effects, measurement uncertainty, reference-value spread, and unknown ν together preclude quoting cp more precisely than a few per cent for an unknown coin.These limits do not threaten the separation of silver from the non-silver groups measured here.Source: D. Why no threshold on cp · p. 13

E. Comparison with prior work

  • Unlike frequency-library methods and, this method measures a material constant conditioned on assumed Poisson’s ratio and has no supported-coin coverage list.Its narrower claim relative to ultrasonic pulse-velocity testing is bulk elastic measurement from a tap and microphone without contact transducer, couplant, or dedicated instrument.Source: E. Comparison with prior work · pp. 13–14

F. Limitations

The method separates known specimens and recovers wave-speed groups, with six struck copper rounds providing its strongest independent validation. Its claims remain bounded by unverified compositions, measurement assumptions, unresolved design effects, and the absence of blind classification.

F. Limitations

  • Without specimens relabelled by a third party and classified blind, the study demonstrates separation of known specimens, not a validated classifier.Sensitivity, specificity, and false-positive rates therefore cannot be quoted.Source: F. Limitations · p. 14
  • Tungsten defeats the density rule, but acoustic rejection remains predicted rather than demonstrated because neither specimen produced a reproducibly detectable resonance.Injection tests do not establish that the specimens lack a resonance; sufficiently lossy substitutes may yield no acoustic reading, and the detection boundary is empirical.Source: F. Limitations · p. 14
  • The apparent residual design dependence is a few per cent on one deeply relieved silver design, with relief depth unmeasured and design confounded with die.The proposed mechanism is consistent with the observation but not established.Source: F. Limitations · p. 14
  • Three specimens have displaced mass measured only at 0.1 g, two five-ounce copper rounds lack displacement measurements, and the finer-resolution 0.5% systematic floor lacks independent calibration.The floor rests on one machined disc and fine-silver density excess, which may be common bias rather than scatter.Source: F. Limitations · p. 14
  • The readout depends on an assumed Poisson’s ratio, and its per-specimen uncertainties exclude that term, relief-model error, displacement bias beyond the floor, and β-transfer error.β is an empirical coefficient fitted from five discs.Source: F. Limitations · p. 14
  • The unverified rounds’ steel-like and brass-like labels are inferences from measured properties; their compositions have not been independently established.Confirmation requires independent bulk analysis, such as a chemical assay or cross-section spectroscopy.Source: F. Limitations · p. 14
  • The unverified rounds’ 1.5% to 3.3% wave-speed uncertainty is dominated by diameters read to 0.5 mm, while their 2% to 5% deficit against plain-carbon steel is of similar order.Displaced volume couples the acoustic readout to density; the unmatched-frequency comparison avoids that coupling but is uncontrolled for thickness or relief.Source: F. Limitations · p. 14

VI. CONCLUSION

  • 119.1 ± 6.7 GPa was recovered for six struck copper rounds, within the accepted 110 GPa to 128 GPa range, but the groups do not resolve all six as distinct composition classes.Five of six groups fall inside or at the edge of accepted modulus ranges; the single brass-like round falls clearly below.Source: §VI · p. 14

APPENDIX A FREE-PLATE COEFFICIENTS BY DIRECT SOLUTION

The appendix derives free-plate coefficients by solving the eigenvalue problem directly, then records analysis choices and checks supporting the reported calibration. Solver comparisons agree closely with tabulated coefficients, while provenance checks show the calibration is insensitive to doublet treatment.

APPENDIX A FREE-PLATE COEFFICIENTS BY DIRECT SOLUTION

  • Directly solving the eigenvalue problem supplies coefficients for specimens spanning ν = 0.28 to 0.37, beyond the two Poisson’s ratios tabulated by Leissa.The free-edge conditions are zero radial bending moment and zero Kelvin–Kirchhoff shear; determinant roots yield λ2.Source: Appendix A · p. 15
  • Reversing the relative sign between VJ and VI leaves n = 0 and n = 1 roots correct but displaces every n ≥ 2 root, so Table IX serves as a check.Two independent implementations agree.Source: Appendix A · p. 15, Table IX · p. 15
  • Agreement is within 0.04% against Table 2.6 and 0.09% against Table 2.5 on the six modes tabulated at both Poisson’s ratios.The Table 2.5 comparison is made at ν = 1/3, which reproduces its values more closely than ν = 0.33.Source: Appendix A · p. 15, Table IX · p. 15

APPENDIX B ANALYSIS PROVENANCE

  • Changing the minimum peak separation from 30 Hz to 8 Hz changed assigned fundamentals for 91 of 214 shared taps, but calibration C shifted by less than 0.02%.The earlier setting merged doublets narrower than 30 Hz; recovered modulus changed by less than 0.03 GPa.Source: Appendix B · p. 15, Table X · p. 15
  • The five aluminium fundamentals reported in Table VI come from the regenerated file, and rerunning the final program on original recordings reproduces that file exactly.The calibration is insensitive to the doublet-treatment choice.Source: Appendix B · p. 15, Table X · p. 15
  • For calibration discs, specimen-level f1 is the dominant-cluster mean, not the lowest cluster.On one disc, the lowest cluster was a single stray tap 0.2% below a population of thirty-two.Source: Appendix B · p. 15
  • That stray tap would bias the specimen’s fundamental by 0.2%, comparable to the entire between-session reproducibility budget.The comparison concerns the calibration estimator and session reproducibility.Source: Appendix B · p. 16
  • Specimen identity is joined exactly through the take log and recording number; five scrap or dud takes passed quality gates but excluding them shifts no fundamental by more than 0.02%.One disputed log entry is assigned by its spectrum; frequency alone would not distinguish some specimens.Source: Appendix B · p. 16
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