Exploring Fourier methods with beer bottles

David Kordahl; Emma Foster

REVIEW 2 major objections 5 minor 12 references

Exploring Fourier methods with beer bottles

T0 review · 2 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read A beer bottle's resonance can be fully mapped from a few seconds of chirp sound using Fourier methods, yielding the same oscillator parameters as slow point-by-point tone sweeps.

desk verdict A solid, honest teaching paper that demonstrates three FFT-based ways to extract a resonator's Green's function from a beer bottle, with a useful correction to the earlier soda-can normalization; the main weakness is an untested linear-superposition assumption. read the letter →

arxiv 2512.12312 v1 pith:7VB5BZ7W submitted 2025-12-13 physics.ed-ph physics.app-phphysics.class-ph

David Kordahl , Emma Foster This is my paper · ORCID

classification physics.ed-phphysics.app-phphysics.class-ph

keywords drivendampedoscillatorGreen'sfunctionFouriertransformchirpsignalHelmholtzresonanceFFTmagnitudeandphaseconvolutiontheoremundergraduatelaboratory

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper shows that the acoustical resonance of a beer bottle is accurately described as a driven damped harmonic oscillator, with a complex Green's function G(ω) that encodes amplitude and phase response. The authors demonstrate that this full frequency response can be extracted from a short chirp signal in two ways: one using only FFT magnitudes, and one also using FFT phase via a numerical convolution. The two chirp methods recover resonance frequency, damping, and coupling parameters consistent with traditional pure-tone measurements, but in seconds rather than minutes. The practical payoff is that students can learn Fourier analysis and system identification with simple, inexpensive equipment and a familiar resonator.

What carries the argument

The central object is the Green's function of the driven damped oscillator, Eq. (7), which connects the speaker pressure to the bottle pressure in the frequency domain. The methodological key machinery is the Fourier transform and the convolution theorem: the incoherent method uses the squared-magnitude ratio R(ω) of the measured spectra, while the coherent method uses the identity G(ω) = F[p_S(−t) * p_M(t)] / |p̃_S(ω)|² − 1, which lets the full complex response be reconstructed from time-reversed convolution without assuming the oscillator lineshape in advance.

What would settle it

Vary the drive amplitude over a wide range (e.g., a factor of 10) and recompute G(ω) from the same chirp method; for a truly linear system the extracted α, β, and ω₀ should be amplitude-independent. If parameters drift systematically with amplitude, the superposition assumption of Eq. (11) breaks down and the claimed Green's function is not intrinsic.

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Extended reading notes

Core claim

The paper establishes that the beer bottle's microphone signal is well modeled as the linear superposition of a speaker background and a bottle contribution, with the bottle described by the Green's function G(ω) = 2αβω₀ / ((ω₀² − ω²) + 2iβω). Measuring the microphone signal with and without the bottle and taking FFTs yields the ratio R(ω) = |p̃_M(ω)|²/|p̃_S(ω)|², which fits the model to extract α, β, and ω₀ from magnitude data alone. Alternatively, convolving the time-reversed speaker signal with the microphone signal and applying the convolution theorem directly recovers the full complex G(ω). All three methods give consistent parameters (α ≈ 3.0–3.4, β ≈ 10–12 Hz, ω₀ ≈ 1220–1221 rad/s), c

Load-bearing premise

The whole analysis assumes the microphone signal is the linear sum of a speaker contribution and an additive bottle contribution, and that the speaker contribution is identical in the with-bottle and without-bottle recordings; if the bottle scatters sound back into the speaker or otherwise changes the speaker's acoustic load, the extracted G(ω) is an artifact of the subtraction scheme.

Editorial extensions

If this is right

Undergraduate labs can measure a resonator's full frequency response in seconds rather than through laborious pure-tone sweeps, making Fourier methods tangible in mechanics courses.
The same Green's-function extraction procedure applies to any resonator with an additive background, not just beer bottles—soda cans, Helmholtz resonators, or cavities of arbitrary shape.
Because the coherent method reconstructs G(ω) directly from data, it can test whether a resonator actually follows the driven-damped oscillator model, not just fit it.
The methods teach students practical signal processing—FFT binning, windowing, convolution, and phase unwrapping—embedded in an accessible physical system.
Using arbitrary chirp signals rather than pure tones demonstrates that any broadband excitation with sufficient spectral content near resonance can serve as a probe.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The deconvolution step of Eq. (38) is a generic linear-system identification technique: given an input and output signal of any causal LTI system, the complex frequency response is recovered by the same time-reversed convolution, so the paper's method transfers beyond acoustics to electronics, mechanics, or optics with suitable transducers.
A natural classroom extension the authors hint at but do not pursue is measuring a chain of coupled bottles; the Green's function of coupled resonators would show mode splitting, directly illustrating normal modes and avoided crossings via the same chirp-and-FFT pipeline.
The strong dependence of α on microphone distance (noted in Sec. IV) suggests that α is a geometric coupling factor rather than an intrinsic bottle property; a testable corollary is that α should scale with the inverse square of microphone-bottle separation, and the coherent method could verify this without refitting.
Because the incoherent magnitude-only method ignores phase, it remains robust to timing jitter between measurements; this suggests a practical advantage when trigger synchronization is imperfect.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper is a good undergraduate lab exercise, and its main genuinely new point is that the raw P_M/P_S ratio used by Wilkinson et al. cannot be fit by the driven-damped oscillator model, while the inferred P_B/P_S can. The three methods—pure tones, incoherent chirp, and coherent chirp—agree within quoted errors, and the coherent method's model-free deconvolution gives real evidence that the bottle's response has the expected oscillator form. The algebra is clean, the setup is cheap, and the explicit credit to prior work (Wilkinson et al., Lemoult et al.) is appropriate.

The soft spots are real but not fatal. The linear-superposition premise of Eq. 11 is load-bearing: the extraction assumes the microphone signal is the speaker field plus an additive bottle field, and that the speaker field is unchanged when the bottle is present. The paper never directly tests this, and a sealed-bottle control would have been a cheap way to do so. The observed mismatch of P_M/P_S is not a control—it's the model's own prediction. Still, the three-method agreement, including the phase-sensitive coherent method, makes it plausible that the subtraction works at these amplitudes. I'd call this a moderate concern, not a reason to reject.

More minor: data and scripts are only 'available upon request,' which undercuts reproducibility, and Fig. 3 lacks per-point error bars. For a pedagogical paper these are acceptable but worth mentioning. The parameter α is acknowledged to depend on microphone distance, which limits the physical interpretation but not the pedagogical point.

Overall, the paper is honest, clearly written, and internally consistent. The stress-test note's concern is legitimate but does not sink the paper; it just deserves a sentence or a control experiment in any revision. The right audience is instructors of upper-division labs or techniques courses that cover Fourier methods and linear response. It deserves a serious referee, and I'd accept it with minor revisions if the authors can either provide the data online or add a sealed-bottle check.

Would I bring it to reading group? Maybe, if the group cares about undergraduate lab design. Would I cite it? Probably not in my own work, but I'd pass it to a colleague who teaches this material.

Referee Report

2 major / 5 minor

Summary. The paper describes an undergraduate acoustics experiment in which a beer bottle is modeled as a driven, damped harmonic oscillator. The model is introduced in Sec. II (Eqs. 1–10), with the bottle's pressure contribution p_B related to the speaker pressure p_S by a Green's function G(ω). Three experimental methods are then compared: steady-state pure-tone measurements (Sec. III), an FFT-magnitude method using chirp signals (Sec. IV A), and an FFT-phase/deconvolution method (Sec. IV B). The paper reports consistent parameter estimates for the bottle's resonance (α≈3.0–3.4, β≈10.4–11.7 Hz, ω₀≈1220.4–1221.1 rad/s) and argues that Fourier methods allow the full complex G(ω) to be obtained from a few seconds of data.

Significance. If the results hold, this is a genuinely useful contribution to the physics-education literature. The three methods are pedagogically attractive: they connect driven-oscillator theory, Fourier transforms, and experimental data acquisition, and the coherent deconvolution method of Eq. (38) is an elegant and instructive application of the convolution theorem. The internal consistency of the three methods is real evidence that the experimental procedures are not grossly wrong, and the model-free character of Eq. (38) means that the agreement in Fig. 6 is not a pure tautology. The paper is also commendably explicit about equipment costs, fitting procedures, and parameter uncertainties. The main weakness is that all three methods rest on an untested linear-superposition assumption, as detailed below.

major comments (2)

[Secs. II–IV (Eq. 11; Eqs. 21, 30, 38)] The load-bearing premise is Eq. (11), p_M = p_S + p_B, together with the assumption that p_S is the same in the with-bottle and without-bottle recordings. This premise is used by all three extraction methods: the pure-tone fit of Eq. (21), the incoherent chirp ratio of Eq. (30), and the coherent deconvolution of Eq. (38). The authors acknowledge this as a 'contention' (Sec. III) and say 'up to uncontrolled fluctuations' the input is the same (Sec. IV), but they provide no independent test. The observation that P_M/P_S fails the model while P_B/P_S fits is not a control, because that failure is itself predicted by the superposition model. If placing the bottle changes the speaker's acoustic load or introduces non-additive scattering, the extracted G(ω) would be an artifact of the subtraction scheme rather than an intrinsic bottle property. Please add an explicit control: for example, vary
[Sec. IV B, Fig. 6 and Eqs. (38)–(39)] The paper's central validation of the oscillator model is the 'quite an impressive match' in Fig. 6 between the model-free G(ω) from Eq. (38) and the fitted oscillator form. However, the agreement is assessed only visually. There are no error bars or shaded uncertainty bands on the extracted G(ω), no residual plot, and no quantitative goodness-of-fit statistic. Since the quoted parameter uncertainties in Eq. (39) are used to claim consistency with the other methods, the authors should either display residuals or provide a χ²-type comparison, and should estimate how FFT noise and finite chirp bandwidth propagate into the extracted G(ω).

minor comments (5)

[Eqs. (22), (32), (39)] The values of ω₀ are quoted in 'Hz,' but they are angular frequencies (ω=2πf, with the resonance near 194 Hz in ordinary frequency). Please use 'rad/s' or explicitly define the convention.
[Fig. 3 and Sec. III] Individual data points in Fig. 3 appear to have no error bars. Since the quoted parameter uncertainties are said to include both fitting uncertainty and sample variation, a sentence explaining how point-to-point uncertainties were estimated (or why they are omitted) would improve reproducibility.
[Eq. (31)] The approximate relations for α, β, and ω₀ are said to be obtained 'analyzing Eq. 30,' but no derivation is given. As a rough check, for α=3, β=10 s⁻¹, ω₀=1220 rad/s the formula for β gives about 11.5 s⁻¹, i.e. a 15% overestimate. This is acceptable for an initial guess, but the text should state the approximation order and the conditions under which it is valid.
[Sec. IV A] The term 'incoherent' is used for the magnitude-only method, while the coherent method also uses phase. This is fine, but consider defining these terms on first use so that students understand that the measured signals are individually coherent; it is the analysis that discards phase information.
[Supplementary Materials] The paper states that data and scripts are 'available upon request.' If the journal supports it, please provide a permanent repository link or DOI; this would materially strengthen the reproducibility of the experiment.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: fitted parameters are honestly calibrated and the coherent chirp extraction is model-free deconvolution.

full rationale

The paper's derivation chain is self-contained and non-circular. The oscillator model (Eq. 1) and Green's function (Eq. 7) are an assumed physical ansatz with free parameters (α, β, ω0). The pure-tone method fits measured P_B/P_S to Eq. 21, explicitly reporting fit parameters (Eq. 22) and noting they can be estimated from the plot; this is calibration, not a disguised prediction. The incoherent-chirp method likewise fits the measured R(ω) to Eq. 30, which is obtained by inserting the assumed G(ω) into Eq. 26; again the parameters are fitted, with initial estimates derived from features of R(ω). The coherent-chirp method (Eq. 38) derives G(ω) directly from measured time series using only the linear-superposition relation p_M = p_S + p_B (Eq. 11), the definition G = p_B/p_S (Eq. 6), and the convolution theorem; no oscillator form is assumed in the extraction itself, so Fig. 6's agreement with Eq. 7 is genuine evidence rather than a tautology. The paper even notes that using the same datasets for Eq. 32 and Eq. 39 makes consistency unsurprising, correctly declining to present this as independent confirmation. No load-bearing self-citations appear (citations [2], [12], and standard textbooks are external). The only serious threat—the untested assumption that p_S is identical with and without the bottle—is a validity/linearity concern, not an instance of the derivation reducing to its inputs by construction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

Three fitted parameters (α, β, ω₀) carry the model; none is derived from bottle geometry or independent measurements. The linear-superposition axiom (Eq. 11) is the deepest structural input: if the bottle's presence measurably changes the speaker's output or adds non-additive reflections, all extracted G(ω) values are artifacts. No new entities are invented — G(ω) is the standard transfer function from the cited literature. The stationarity and chirp-coverage assumptions are explicit in the text and could be violated at the sweep edges, which the paper acknowledges for the coherent method ('outside the frequency range of the sweep tone... not large enough to produce stable quotients').

free parameters (3)

α (dimensionless speaker–bottle coupling) = 3.0–3.4 (pure tones: 3.4±0.2; incoherent: 3.0±0.4; coherent: 3.0±0.4)
Fit to the height of the P_B/P_S resonance peak (Eq. 21) in pure tones and to R(ω) in chirps (Eq. 30); the paper notes it depends on microphone–bottle distance. Not computed from geometry or independent measurements.
β (damping rate) = 10.4–11.7 Hz (pure tones: 10.4±0.7; incoherent: 11.0±0.8; coherent: 11.7±1.2)
Fit to the width of the resonance peak. Damping is not derived from first principles (e.g., neck geometry or wall losses).
ω₀ (undamped resonance angular frequency) = 1220.4–1221.1 rad/s (≈194 Hz)
Fit to the peak/center of the resonance. The geometric Helmholtz prediction is cited only as background [8] and never used to check the fitted value.

assumptions (5)

domain assumption Linear superposition of acoustic pressures: p_M = p_S + p_B (Eq. 11).
Underlies every extraction in the paper; all three methods subtract or deconvolve the without-bottle recording from the with-bottle recording on this basis. Never independently verified (e.g., by a linearity check).
domain assumption Single-mode driven-damped-oscillator dynamics (Eq. 1).
Postulates one Helmholtz mode with restoring, driving, and damping terms; higher-order harmonics and asymmetric-cavity corrections are explicitly left for future work (Sec. V).
standard math Fourier transform/inverse transform (Eqs. 2–3) and convolution theorem (Eqs. 33–34).
Used without proof; both are standard results in the cited textbooks [3,5].
domain assumption Stationarity of the acoustic environment across the two recordings.
Sec. IV: 'Up to uncontrolled fluctuations, the input signal should be the same for the two time-dependent measurements'; temperature variation is quoted at <1°C. Any drift between recordings injects systematic error into the ratio/deconvolution.
domain assumption Chirp Fourier components near ω₀ are large enough.
Sec. IV intro: the methods 'do not explicitly depend on the form of input signal, as long as the Fourier components close to ω₀ are large enough'. A null in the chirp or apparatus response near resonance would make the extracted G(ω) noise-dominated, as the paper itself notes at the sweep edges.

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Cite this review

Pith. "Pith review of Exploring Fourier methods with beer bottles." pith.science (2026). https://pith.science/paper/7VB5BZ7W

@misc{pith2026251212312,
  author       = {Pith},
  title        = {Pith review of: Exploring Fourier methods with beer bottles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7VB5BZ7W}},
  note         = {Machine review of arXiv:2512.12312}
}

read the original abstract

As anyone who has blown across the mouth of a beer bottle knows, beer bottles have a well-defined fundamental frequency. This paper shows how a beer bottle's acoustical resonance can be modeled as a one-dimensional driven-damped oscillator and includes enough detail to be useful in undergraduate laboratory experiments. While the frequency-domain Green's function of the bottle can be extracted through sequential pure-tone measurements, sufficient data to fit the model's parameters can be collected in just a few seconds when Fourier methods are used.

Figures

Figures reproduced from arXiv: 2512.12312 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** FIG. 3. Measurements obtained from pure tones at different [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4. Measurements corresponding to a 20.0 s chirp signal [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references

[1]

The great beer bottle experiment,

Gary R. Smith and P.D. Loly, “The great beer bottle experiment,”Am. J. Phys.47, 515-518 (1979)

1979
[2]

An undergraduate experiment demonstrating the physics of metamaterials with acoustic waves and soda cans,

James T. Wilkinson, Christopher B. Whitehouse, Rupert F. Oulton, and Sylvain D. Gennaro, “An undergraduate experiment demonstrating the physics of metamaterials with acoustic waves and soda cans,”Am. J. Phys.84, 14–20 (2016)

2016
[3]

7 of Mary L

See Ch. 7 of Mary L. Boas,Mathematical Methods in the Physical Sciences, Third Edition(Wiley, 2006)

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[4]

3 of David J

See Ch. 3 of David J. Griffiths,Introduction to Electro- dynamics, 4th Edition(Cambridge UP, 2017)

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[5]

5 of John R

See Ch. 5 of John R. Taylor,Classical Mechanics(Uni- versity Science Books, 2005)

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[6]

audacityteam.org/

Audacity can be downloaded for free athttps://www. audacityteam.org/
[7]

Alexander J

Hermann von Helmholtz, trans. Alexander J. Ellis.On the Sensations of Tone as a Physiological Basis for the Theory of Music. (Longmans, Green, and Co., 1885; reprintedDover, 1954)

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[8]

phys.unsw.edu.au/jw/Helmholtz.html

See the Helmholtz resonator discussion athttp://newt. phys.unsw.edu.au/jw/Helmholtz.html

Show all 12 references

[9]

The Green of Green Functions,

For historical background, see Lawrie Challis, “The Green of Green Functions,”Physics Today56 (12), 41–46 (2003)

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[10]

Data Fitting in Python Part II: Gaussian & Lorentzian & Voigt Lineshapes, Deconvoluting Peaks, and Fitting Residu- als,

A tutorial has been written by Emily Grace Ripka, “Data Fitting in Python Part II: Gaussian & Lorentzian & Voigt Lineshapes, Deconvoluting Peaks, and Fitting Residu- als,” athttp://www.emilygraceripka.com/blog/16
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See Ch. 11, Steven W. Smith,Digital Signal Processing: A Practical Guide for Engineers and Scientists(Newnes, 2002)

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[12]

Acoustic Resonators for Far-Field Control of Sound on a Subwavelength Scale,

This method was suggested in the Supplemental Material of Fabrice Lemoult, Mathias Fink, and Geoffroy Lerosey, “Acoustic Resonators for Far-Field Control of Sound on a Subwavelength Scale,”Phys. Rev. Lett.107, 064301 (2011)

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

The great beer bottle experiment,

Gary R. Smith and P.D. Loly, “The great beer bottle experiment,”Am. J. Phys.47, 515-518 (1979)

1979

[2] [2]

An undergraduate experiment demonstrating the physics of metamaterials with acoustic waves and soda cans,

James T. Wilkinson, Christopher B. Whitehouse, Rupert F. Oulton, and Sylvain D. Gennaro, “An undergraduate experiment demonstrating the physics of metamaterials with acoustic waves and soda cans,”Am. J. Phys.84, 14–20 (2016)

2016

[3] [3]

7 of Mary L

See Ch. 7 of Mary L. Boas,Mathematical Methods in the Physical Sciences, Third Edition(Wiley, 2006)

2006

[4] [4]

3 of David J

See Ch. 3 of David J. Griffiths,Introduction to Electro- dynamics, 4th Edition(Cambridge UP, 2017)

2017

[5] [5]

5 of John R

See Ch. 5 of John R. Taylor,Classical Mechanics(Uni- versity Science Books, 2005)

2005

[6] [6]

audacityteam.org/

Audacity can be downloaded for free athttps://www. audacityteam.org/

[7] [7]

Alexander J

Hermann von Helmholtz, trans. Alexander J. Ellis.On the Sensations of Tone as a Physiological Basis for the Theory of Music. (Longmans, Green, and Co., 1885; reprintedDover, 1954)

1954

[8] [8]

phys.unsw.edu.au/jw/Helmholtz.html

See the Helmholtz resonator discussion athttp://newt. phys.unsw.edu.au/jw/Helmholtz.html