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# The Brown Note
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# Browntone: Vibroacoustics of Fluid-Filled Soft Shells
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Can airborne sound make you soil yourself? No — and the mechanical-vs-airborne asymmetry is the interesting part.
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This repository contains the analytical framework, source code, and manuscript drafts for a research programme investigating the low-frequency vibroacoustics of fluid-filled viscoelastic shells — with applications ranging from abdominal infrasound coupling to non-destructive produce testing and spectral identifiability theory.
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The abdomen does have a real low-frequency flexural resonance near 4 Hz. The problem for the "brown note" is not resonance; it is coupling. The airborne path couples so weakly to that mode that the real story is the asymmetry between airborne forcing and mechanical vibration — and that turns out to be a useful problem in shell mechanics, vibration, and inverse theory.
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The programme is **purely analytical at present**. A phantom validation protocol has been designed but not yet executed; see [docs/](docs/).
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**Purely analytical at present.** A phantom validation protocol has been designed but not yet executed; see [docs/](docs/).
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## Motivation
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The same shell model that explains why sound cannot shake your insides also explains when you can, and cannot, invert a resonance spectrum to recover the parameters that generated it.
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The human abdomen possesses a real flexural resonance near 4 Hz. The long-standing conjecture — sometimes called the *brown note* — is that sufficiently intense airborne infrasound could exploit this resonance to induce involuntary gastrointestinal effects. We show that the resonance is genuine but the airborne coupling to it is not: at 120 dB SPL the energy-consistent wall displacement is only 0.028 μm, well below the 0.5–2.0 μm range associated with PIEZO-channel mechanotransduction. Whole-body mechanical vibration, by contrast, couples roughly 3.3 × 10⁴ times more effectively (SDOF upper bound; ~1.6 × 10⁴ with modal participation Γ₂ ≈ 0.48).
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## The Short Version
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The interesting result is not the debunking itself but the asymmetry it reveals — and the observation that the same oblate-spheroidal shell model that explains *why* sound cannot shake your insides also determines *when* a resonance spectrum can or cannot identify the parameters that produced it.
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At 120 dB SPL, the energy-consistent airborne displacement of the whole abdominal wall is only 0.028 μm. That sits below the 0.5-2.0 μm cellular activation range usually associated with PIEZO mechanotransduction. The resonance is real; the whole-abdomen airborne coupling to it is not.
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## Principal Papers
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A second way to say the same thing is that H_mech/H_air ≈ 6.5 × 10^6. Paper 1 resolves the mechanical-vs-airborne asymmetry for whole-abdomen coupling, while Paper 2 shows that localised gas pockets are the notable exception.
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### [Paper 1 — The Brown Note](papers/paper1-brown-note/)
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*Targeting Journal of Sound and Vibration*
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Paper 7 is a restricted scalar inversion that estimates effective rind stiffness from a watermelon tap tone. That forward problem is exactly what motivates the broader identifiability question taken up in Papers 8-10.
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Paper 1 resolves the mechanical-vs-airborne coupling asymmetry from first principles. At the n = 2 flexural resonance, ka ≈ 0.01: the acoustic wavelength dwarfs the body, the impedance mismatch is severe, and the incident field barely drives the mode. Mechanical excitation through the body wall suffers none of these penalties. The conclusion is not that abdominal resonance is fictional; it is that the resonance is real while the airborne path to it is negligible.
If you read only one paper, read Paper 1. If you want to see why the framework is useful beyond the original question, read Paper 7.
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Paper 7 applies the same shell framework to a problem with clearer practical value: mapping a watermelon's tap-tone frequency to its effective rind stiffness. The claim is deliberately narrow — inference to eating ripeness requires cultivar-specific calibration not yet demonstrated — but the forward model works well enough to make the inverse question unavoidable: under what conditions does a resonance spectrum actually identify the parameters of interest? That question motivates Papers 8–10.
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### [Paper 1: The Brown Note](papers/paper1-brown-note/) — targeting *Journal of Sound and Vibration*
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##Supporting Analyses (Papers 2–6)
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This is the paper that answers the question. It starts from a simple empirical asymmetry: whole-body vibration in the 4-8 Hz range can produce gastrointestinal effects, while airborne sound at the same frequencies does not. The model explains that asymmetry from first principles.
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Papers 2–6 are not independent contributions; they probe the same shell framework across complementary organs, forcing conditions, and physiological scenarios.
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At the n = 2 flexural resonance, ka is only about 0.01. The wavelength is enormous compared with the body, the air-tissue mismatch is severe, and the incident acoustic field barely drives the mode at all. Mechanical excitation transmitted through the body wall does not suffer those penalties. The result is not that abdominal resonance is fictional; it is that the resonance is real while the airborne coupling is not.
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| Paper | Focus | Contribution |
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|-------|-------|-------------|
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|[P2](papers/paper2-gas-pockets/)| Gas pocket transduction | Tissue-constrained bubbles couple far more strongly than the whole cavity |
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|[P3](papers/paper3-scaling-laws/)| Cross-species scaling | The low-frequency shell mechanics are not uniquely human |
|[P6](papers/paper6-sub-bass/)| Sub-bass perception | Structural transmission dominates the airborne path at concert levels |
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### [Paper 7: The Watermelon Thump Test](papers/paper7-watermelon/) — targeting *Postharvest Biology and Technology*
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##Identifiability Theory (Papers 8–10)
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This is the unexpected bridge paper. People really do tap watermelons and listen for a deep, resonant thump, but the practice has mostly lived as anecdote and market lore. Paper 7 turns that intuition into a quantitative shell model that maps tap-tone frequency to effective rind stiffness.
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The theoretical contribution of the programme rests on four formal results:
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That is a narrower claim than "ripeness prediction". Inference to eating ripeness requires cultivar-specific calibration not yet demonstrated. What Paper 7 contributes is a forward model that works well enough to make the inverse question unavoidable: when does a resonance spectrum actually identify the parameters you care about?
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1.**Rank deficiency under scalar geometric reduction.** Equivalent-radius formulations reduce the Jacobian's column space and can render spectral inversion structurally ill-posed ([P8](papers/paper8-kac/), targeting a suitable inverse-problems venue).
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2.**Identifiability lifting by oblate asphericity.** Breaking spherical symmetry in the oblate direction restores local rank and improves conditioning by ten orders of magnitude ([P8](papers/paper8-kac/)–[P10](papers/paper10-capstone/)).
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3.**Near-spherical conditioning asymptotics.** As eccentricity ε → 0, the smallest singular value σ₃ vanishes as λ₁ε² + O(ε⁴); the finite floor reported in earlier Ritz calculations was a discretisation artefact ([P9](papers/paper9-lifting-theorem/)–[P10](papers/paper10-capstone/)).
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4.**Forward adequacy does not imply inverse adequacy.** A model can predict resonance frequencies to within 10% yet remain catastrophically ill-conditioned for parameter recovery ([P10](papers/paper10-capstone/)).
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## Supporting Analyses
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[Paper 9](papers/paper9-lifting-theorem/) establishes that comparable prolate perturbations do not produce the same lifting — the mechanism is geometry-selective. [Paper 10](papers/paper10-capstone/) contains the formal proofs for the axisymmetric case; extensions to non-axisymmetric geometries and experimental validation are the principal next steps.
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Papers 2-6 are supporting analyses, not five unrelated grand claims. They test the same framework across related organs, forcing conditions, and physiological scenarios.
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## Selected Quantitative Results
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| Paper | Focus | What it adds |
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|---|---|---|
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|[P2 Gas Pockets](papers/paper2-gas-pockets/)| Local gas inclusions | Tissue-constrained bubbles can couple much more strongly than the whole abdominal cavity. |
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|[P3 Scaling Laws](papers/paper3-scaling-laws/)| Cross-species scaling | The low-frequency shell mechanics are not uniquely human. |
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|[P4 Bladder Resonance](papers/paper4-bladder/)| Organ-specific variation | The same framework predicts a different resonance landscape for the bladder. |
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|[P5 Borborygmi](papers/paper5-borborygmi/)| Gut sounds | Bubble-shell acoustics can account for clinically realistic stomach-growl frequencies. |
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|[P6 Sub-Bass Coupling](papers/paper6-sub-bass/)| Airborne vs structural bass | Reinforces that the floor and seat matter far more than the air. |
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## Formal Results
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The theoretical spine of the programme is compact enough to state plainly:
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1.**Rank deficiency of equivalent-radius models.** Sphere-like equivalent-radius formulations are badly ill-conditioned for spectral inversion (P8, targeting a suitable inverse-problems venue).
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2.**Identifiability lifting by oblate asphericity.** Breaking spherical symmetry in the oblate direction lifts identifiability and improves conditioning (P8-P10).
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3.**Near-spherical conditioning asymptotics.** As spherical symmetry is restored, conditioning worsens sharply; the apparent finite floor in earlier low-order Ritz calculations was a discretisation artefact, and the correct limit is κ → ∞ as ε → 0 (P9-P10).
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4.**Forward adequacy ≠ inverse adequacy.** A model can predict resonance frequencies well and still fail as a parameter-identification tool; P7 is the motivating example, P8-P10 make the point formal.
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P9 adds the cross-check that comparable prolate perturbations do not produce the same lifting. P10 currently contains the formal results for the axisymmetric case; extension to non-axisymmetric geometries and experimental validation are the next steps.
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## Key Results
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| Quantity | Value | Paper |
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|----------|-------|-------|
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| Quantity | Value | Source |
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|----------|-------|--------|
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| Flexural frequency f₂ (n = 2) | 3.95 Hz | P1 |
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| Breathing mode (n = 0) | 2490 Hz | P1 |
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| Airborne displacement ξ_air at 120 dB | 0.028 μm | P1 |
-**An analytical programme, not an experimental one.**The papers and code are modelling work; an experimental validation protocol exists, but it has not yet been executed.
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-**A rind-stiffness framework, not a ripeness oracle.** Paper 7 maps tap-tone frequency to effective rind stiffness; inference to eating ripeness requires cultivar-specific calibration not yet demonstrated.
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-**A mostly finished arc, not a closed book.**Papers 1-9 contain the current core results. Paper 10 is the synthesis still being refined.
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-**Latest Paper 10 draft.**[papers/paper10-capstone/drafts/draft_2026-04-04_1400.pdf](papers/paper10-capstone/drafts/draft_2026-04-04_1400.pdf)
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-**Analytical, not experimental.**All results derive from shell-theoretic models. A phantom validation protocol exists ([docs/](docs/)) but has not been executed.
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-**Rind stiffness, not ripeness.** Paper 7 recovers effective stiffness from tap-tone data; the link to consumer-relevant ripeness requires cultivar calibration not yet performed.
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-**Papers 1–9 are submission-ready drafts.** Paper 10 is under active revision.
The core model treats the abdomen as a fluid-filled viscoelastic oblate spheroidal shell. In the forward problem, an equivalent-radius formulation captures the dominant low-frequency behaviour. In the inverse problem, the full oblate geometry matters because asphericity is exactly what breaks the spectral degeneracy. Flexural modes (n ≥ 2) govern the infrasonic response; the breathing mode (n = 0) sits around 2490 Hz and is not the low-frequency story.
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### Quick Start
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## Getting Started
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```bash
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pip install -e .[dev]
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python -m pytest tests/ -v
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```
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```python
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from src.analytical.natural_frequency_v2 import (
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AbdominalModelV2,
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flexural_mode_frequencies_v2,
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)
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from src.analytical.natural_frequency_v2 import AbdominalModelV2, flexural_mode_frequencies_v2
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from src.analytical.energy_budget import self_consistent_displacement
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from src.analytical.mechanical_coupling import compare_airborne_vs_mechanical
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