Methodology

In short: physical specifications come from the issuing mints; the expected resonance ranges are computed by us from first-principles acoustic physics — the mint's mass and diameter, published materials constants, and classical free-plate theory — and published as guidance bands rather than exact values. We aggregate test values to build an internal sample database for future adaptive verification.

Weight, diameter, thickness, and fineness for each preset coin are taken from the issuing mint's published data (for example the U.S. Mint, The Royal Mint, Royal Canadian Mint, Perth Mint, and the Austrian Mint). Each per-coin guide page links the specific source it uses. Where a mint does not publish a figure (older classics, private-mint rounds), we fall back to the Numista catalogue or the manufacturer's product specification, and note the lower confidence.

Every reference center in the app is computed from this model; separately, we're aggregating real-world test values into our sample database for a secondary, standard-distribution-based lookup as a secondary check (future feature). A struck coin vibrates as a free circular plate. Its audible peaks are the plate's flexural modes with zero nodal circles and 2, 3, and 4 nodal diameters — what we label c0d2, c0d3, and c0d4. Classical plate theory gives each mode's frequency in closed form:

f = (λ² / (2π·a²)) · √( D / (ρ·h) ),   D = E·h³ / (12·(1 − ν²))

  a = plate radius     h = thickness      ρ = density
  E = Young's modulus  ν = Poisson ratio  D = flexural rigidity

The dimensionless frequency parameters λ² for a free circular plate are tabulated in the classical literature — λ² ≈ 5.25, 12.2, and 21.6 for the c0d2 / c0d3 / c0d4 modes (Leissa, 1969; originally Colwell & Hardy, 1937). These are geometry-only and give the fixed modal spacing of roughly 1 : 2.3 : 3.9 that our non-coin filter checks for.

The material term uses standard published constants (treat Young's modulus as ±10%; alloy density dominates for gold):

Elastic constants from standard materials references (ESPI Metals; MakeItFrom; material-properties.org); alloy densities from mint specifications. Because a coin's average disc thickness is less than its rim-to-rim caliper reading, the model uses an effective thickness, h_eff = m / (ρ · π·d²/4), derived from mass and density. A fixed rim/relief correction of ×0.967 (−3.3%) accounts for the raised rim and design relief lowering the fundamental slightly below the ideal flat-disc value. The higher modes start from the theoretical λ² ratios but get a thick-plate (finite-thickness shear) correction: real coins are not infinitely thin, so their higher modes fall below the ideal ratios by an amount that grows with the mode and with the coin's thickness-to-radius ratio (h/a)² — computed from each coin's own geometry (a single order-0.1 shear coefficient, consistent with Mindlin plate theory). *The annealed-fine-gold modulus is the least-certain input — see §3.

A model is only as good as its inputs. The dominant uncertainty is the elastic modulus, which we treat as roughly ±10%; because frequency goes as √E, that is about ±5% on each center, best for silver (whose constants are well established) and largest for .9999 gold, whose annealed modulus is genuinely uncertain in the literature. The mode ratios are essentially exact — a geometry-only 1 : 2.33 : 4.11 — which is what our non-coin filter checks. Independent measurements corroborate the model: Lawrence Livermore (below) reports circulating-coin frequencies within about 1 kHz of the same free-plate calculation.

Two consequences we state plainly. First, these are guidance bands, not a measured database — the physics is precise enough to catch gross wrong-metal fakes (which shift frequencies by tens of percent) but not to replace a measured reference for a good same-alloy counterfeit, so calibrating your own genuine coin remains the most accurate path. Second, we widen the tolerance for the least-certain groups — .9999 gold in particular — so a genuine specimen still falls inside its band; a direct measurement of a known-good coin is the anchor we use to tighten those groups over time.

The same free-plate model, and acoustic coin authentication in general, are well established in the peer-reviewed and institutional literature:

• Lawrence Livermore National Laboratory, Resonant Acoustic Characterization of Coins (LLNL-TR-829101) — applies the free-edge flat-disc model with effective thickness to circulating coins, reporting measured frequencies within ~1 kHz of the model.
• T.-P. Dao et al., "Development of an Electromagnetic Acoustic Transducer System for Coin Classification," Sensors 22(23):9055 (2022) — uses the identical λ²/(2πr²)·√(D/ρh) formulation to classify genuine vs. counterfeit coins.
• "The music of gold: can gold counterfeited coins be detected by ear?" — thin-plate (Kirchhoff) analysis of a gold coin's fundamental modes, with a worked numerical example.

Reference centers are rounded, and each mode is published as a guidance range (a ± band). Rather than adopt any inherited tolerance, we derive each band from first principles as the expected spread of a genuine specimen's ring, combined in quadrature:

band = √( sensitivity² + capture² + split² + class² + model² )

• Sensitivity — how much f = h·a⁻²·√(E/ρ) moves under real manufacturing tolerance (diameter ±0.1 mm, weight ±0.5%) and a ±15 K temperature swing. For modern bullion this is under 1% — the coin itself is made too precisely to vary much.
• Capture — reproducibility of a phone-mic strike (microphone, holding/damping, strike point, room): ≈4%. This dominates the band. It is our own engineering allowance for the measurement chain, which our opt-in sample data will refine over time.
• Split — an extra allowance on the fundamental c0d2 mode, which is most prone to specimen-to-specimen splitting from minting asymmetry (documented in LLNL-TR-829101): ≈3% on c0d2 only.
• Class — a documented widening for coins that genuinely vary more: circulated 90% classics (wear removes mass and edges) and issues struck at different thicknesses across years.
• Model — the center-uncertainty from §3, propagated per metal from the published spread in its elastic modulus (since f ∝ √E, a ±E range gives ±½ that on the center): ≈4% for silver, ≈5% for 22k gold, and largest for .9999 gold (≈7.5%, E known only to ~±15%) until a measured anchor tightens it.

So a pristine silver bullion coin lands near ±6–7%, a .9999 gold coin near ±8–9%, and a worn 90% classic near ±11% — not because we assigned those numbers, but because that is what the physics of a genuine specimen's spread, plus the phone-capture scatter and the model's own uncertainty, works out to. Every band is adjustable per peak inside the app. The ping/ring test itself is a long-established, well-known method, and the mode notation (c0d2/c0d3/c0d4) is standard physics nomenclature. See the algorithm page for how the measurement works.

• A. W. Leissa, Vibration of Plates, NASA SP-160. U.S. Government Printing Office, 1969 (public domain).
• R. C. Colwell and H. C. Hardy, "The frequencies and nodal systems of circular plates," Philosophical Magazine, ser. 7, 24(165):1041–1055, 1937.
• Lawrence Livermore National Laboratory, Resonant Acoustic Characterization of Coins, LLNL-TR-829101.
• T.-P. Dao et al., "Development of an Electromagnetic Acoustic Transducer System for Coin Classification," Sensors 22(23):9055, 2022. PMC9740311
• Materials data: ESPI Metals; MakeItFrom; material-properties.org.

Coin Pinger is an independent project and is not affiliated with, endorsed by, or sponsored by any mint, refiner, or other coin-testing application.