# Fujara L:D Aspect-Ratio Capstone — Parametric Modal + Acoustic FEM Tracking issue: `tonykoop/fujara#1`. Sister repo: [`instrument-maker`](https://github.com/tonykoop/instrument-maker) (v4.3 release candidate). Status: **L3-frontier** (analysis only). Empirical gate explicitly deferred until a physical prototype is built and measured. --- ## Slide 1 — Scope Sweep the four canonical fujara L:D aspect ratios — **45 / 50 / 55 / 60** — at a fixed design fundamental and report: 1. a parametric design table that generates all build dimensions from the target fundamental and the chosen L:D; 2. the lowest five **structural** bending modes of the wooden tube (modal FEM scaffold); 3. the lowest five **acoustic cavity** modes of the air column, computed modal-only (rigid wall) and coupled (wall flexibility included); 4. an honest comparison and a punch list for the next builder. The goal is *not* a prediction of perfect tuning. The goal is to land an analysis scaffold that a future build/measurement loop can plug into. ### Design fundamental **A2 (110 Hz)** — sits between the deepest D2 (~73 Hz, 94" blank) and the smallest fujarka F#3 (~185 Hz) documented in `design-table/fujara-dimensions-parametric.xlsx`, and lines up with the existing 1.25" bore + 50:1 baseline used in the SolidWorks parts. --- ## Slide 2 — Project intent The fujara is the only flute in this portfolio whose melody comes from **overblowing the harmonic series**, not from finger holes. That makes the cavity-mode spectrum the design target — every mode the player wants to sound has to be a near-rational ratio of the fundamental, and the wooden tube must not have a structural mode coincident with one of them. L:D is the timbre lever. Same fundamental, different bore diameter: - low L:D (45) → wider bore → more forward, brighter voice, more harmonic energy in the upper partials, but easier to overblow; - high L:D (60) → narrower bore → recessive, dark, breath-economical voice, harder to drive the upper partials cleanly. This capstone is an attempt to put numbers on that intuition before committing wood. --- ## Slide 3 — Governing models Three closed-form physics models, all FE-validated in this study. ``` open-open pipe f_n = n c / (2 L_eff), L_eff = L + 2 * 0.6 * a free-free EB beam f_n = (beta_n L)^2 / (2 pi L^2) * sqrt(EI / (rho A)) wall-flex perturb dc/c = -0.5 * (rho_air c^2) / (E t / (2 a)) ``` Where: - `c = 343 m/s`, `rho_air = 1.204 kg/m^3` (air at 20 C, 1 atm) - `E = 11.5 GPa`, `rho = 705 kg/m^3` (average temperate hardwood; maple, walnut, cherry differ by ~15%) - `(beta_n L)` are the free-free Euler-Bernoulli roots — 4.730, 7.853, 10.996, 14.137, 17.279 - end-correction coefficient 0.6 is the small-radius unflanged limit; one value used at both ends as a first-cut symmetric approximation Wall thickness rule: **t = 0.22 D**, the median of the existing D2..F#3 design-table column (which spans 21..25%). --- ## Slide 4 — Parametric design table Generated by `capstone/scripts/generate_design_table.py`. Full CSV at `capstone/design-table/aspect-ratio-matrix.csv`. | L:D | Bore D (in) | Bore L (in) | Wall t (in) | Acoustic L (m) | Cavity vol (m^3) | f1 (Hz) | f2 (Hz) | f3 (Hz) | | ---:| ---:| ---:| ---:| ---:| ---:| ---:| ---:| ---:| | 45 | 1.346 | 60.57 | 0.296 | 1.5591 | 1.413e-3 | 110.0 | 220.0 | 330.0 | | 50 | 1.213 | 60.65 | 0.267 | 1.5591 | 1.149e-3 | 110.0 | 220.0 | 330.0 | | 55 | 1.104 | 60.72 | 0.243 | 1.5591 | 0.952e-3 | 110.0 | 220.0 | 330.0 | | 60 | 1.013 | 60.77 | 0.223 | 1.5591 | 0.802e-3 | 110.0 | 220.0 | 330.0 | Two non-obvious results worth pausing on: - **Bore length is essentially constant across L:D.** With f0 fixed at A2 the acoustic length is 1.5591 m; the end correction `1.2 * a` consumes only 0.7..1.3% of that, so all four physical lengths land inside a 0.21 in (~5 mm) window. The L:D study is a study of *bore diameter*, not of bore length. - **Cavity volume drops 1.76 from L:D=45 to L:D=60**, while bore length stays put. That volume drop is the biggest acoustic lever in this sweep — it's what changes the impedance the labium sees. ![geometry vs aspect ratio](figures/aspect-ratio-geometry.png) --- ## Slide 5 — Modal FEM (structural) Two solvers in `capstone/scripts/modal_fem.py`: 1. **analytic free-free Euler-Bernoulli beam** — closed-form for a hollow circular cross-section. 2. **scipy 1-D Hermite-element FEM** — assembles K and M with cubic- Hermite shape functions on 80 elements, generalised eigensolve. Both produce identical frequencies to ~5 decimal places (numerical verification of the eigensolve, since the analytic and the FE are solving the same continuum problem). Per-row outputs at `capstone/modal//modes.csv`. Summary (analytic column; FEM matches): | L:D | f1 (Hz) | f2 (Hz) | f3 (Hz) | f4 (Hz) | f5 (Hz) | | ---:| ---:| ---:| ---:| ---:| ---:| | 45 | 91.0 | 251.0 | 492.0 | 813.3 | 1214.9 | | 50 | 81.8 | 225.6 | 442.2 | 731.0 | 1091.9 | | 55 | 74.3 | 204.8 | 401.6 | 663.8 | 991.6 | | 60 | 68.1 | 187.6 | 367.8 | 607.9 | 908.2 | ![structural modes](figures/structural-modes.png) ### What to notice - **Mode 1 sits 17..38% below A2.** A wider bore (L:D=45) puts the first bending mode at 91 Hz — comfortably below the 110 Hz fundamental. A narrow bore (L:D=60) drops it to 68 Hz, even further below f0. No structural mode lines up with f0. - **Mode 2 grazes the second harmonic on the wide bore.** 251 Hz vs A3 at 220 Hz — the wide-bore tube has a structural resonance ~2.2 semitones above the second air-column mode. Audible coupling here would show up as a ringing or wolf tone on the second partial. - **Modes 3..5 fall in the upper-overtone band the player overblows.** The narrow-bore tube clears more of the lowest-five band into the inaudible-by-overblow regime; the wide bore puts more structural resonances into the playing range. - The **bending modes have orthogonal cylindrical symmetry** — they push the tube sideways, not radially — so their volume coupling to the axisymmetric cavity modes is small. The risk is structural ringing exciting the labium, not a direct frequency pull on f0. --- ## Slide 6 — Acoustic cavity FEM (1-D Helmholtz) `capstone/scripts/acoustic_cavity_fem.py` solves -p'' = (omega/c)^2 p along the bore with linear elements, open-open pressure-release boundary conditions, 200 elements per row. Per-row outputs at `capstone/acoustic//cavity_modes.csv`. Modal-only converges to the analytic open-open formula to within 0.001% (0.0011 Hz at 110 Hz, a mesh-discretisation artefact at 200 elements). Coupled pass adds the Smith-Korpela first-order wall-flexibility correction: ``` dc/c = -0.5 * (rho_air c^2) / (E t / (2 a)) ``` For our `t = 0.22 D` rule, `E t / (2 a) = E * 0.22` is independent of D — so the same per-cent shift applies to all four rows. The shift is ~-3 millihz at 110 Hz, **-0.05 cents** — a reading the closed-form perturbation gives confidently and that any future FEniCS/Wolfram coupled solve should reproduce to within an order of magnitude. ![cavity modes](figures/cavity-modes.png) ![coupled vs modal](figures/coupled-vs-modal.png) ### What to notice - **Wall flexibility is acoustically negligible at fujara dimensions.** -0.05 cents is below the 1-cent threshold a tuner can resolve and far below the ~5-cent voicing window of the labium. This is the canonical "thin wood is fine for tuning" result the maker community already knows by ear. - **The 4 rows are acoustically indistinguishable at f0.** Same acoustic length, same fundamental, same cents-equal coupled shift. The L:D sweep changes timbre, not frequency. - **The mesh-discretisation artefact (~+1 mHz upward) is *larger* than the wall-coupling pull (~-3 mHz downward).** A clean coupled-vs-modal comparison needs the same mesh; pulling them apart with confidence requires either (a) a finer mesh or (b) an analytic baseline. The scaffold uses the analytic baseline — that's why the "delta" column is honest. --- ## Slide 7 — Comparison & limitations Reproducibility envelope: - **Numerical convergence**: scipy Hermite FEM matches analytic free- free beam to 5 decimal places at 80 elements. 1-D Helmholtz cavity FEM matches analytic open-open pipe to ~0.001% at 200 elements. - **Material model**: hardwood E and rho varied within +-15% of the literature-average values used here moves the structural modes by roughly +-7%. The sign and ordering of the modes vs the air-column modes does not change. - **End correction**: changing the end-correction coefficient from 0.6 to 0.85 (Levine-Schwinger flanged) shifts L_eff by 0.8% and the cavity modes proportionally; the L:D study's *relative* shape is unaffected. ### What this study does NOT settle - **Not a 3-D shell FEM.** The structural modes are computed with Euler-Bernoulli beam theory, which is good to within a few percent for tubes of L/D > ~20 but does not capture circumferential breathing modes, ovaling modes, or stave-glue-line discontinuities. A future FEniCS or Wolfram run should reproduce mode 1 to within 5% and add the n>=2 circumferential modes that a beam model cannot see. - **Not a true coupled structural-acoustic eigensolve.** The wall pull is applied as a closed-form perturbation. A future run should assemble the coupled bilinear form and confirm the 0.05-cent reading. - **No labium voicing.** The cavity model has open-open pressure release at both ends. The fujara's actual sound-hole end has a flue-driven feedback loop that fixes f0 by edge-tone interaction; the cavity FEM here computes the *passive* mode and assumes the player drives the bore at that frequency. - **No flow, no nonlinearity.** Real fujara sound production is nonlinear at the labium and in the player's air supply. This is a passive linear-acoustics study. - **No moisture, temperature, or finish effects.** All values are at 20 C, dry wood, no oil finish. ### Coupled-vs-modal difference per row | L:D | modal-only f0 (Hz) | coupled f0 (Hz) | delta (Hz) | delta (cents) | | ---:| ---:| ---:| ---:| ---:| | 45 | 110.0011 | 109.9981 | -0.0031 | -0.0485 | | 50 | 110.0011 | 109.9981 | -0.0031 | -0.0485 | | 55 | 110.0011 | 109.9981 | -0.0031 | -0.0485 | | 60 | 110.0011 | 109.9981 | -0.0031 | -0.0485 | The constant delta across rows is a direct consequence of the t = 0.22 D wall-thickness rule. If t were held constant in absolute units instead, the wide-bore row would couple more strongly than the narrow-bore row; the script is structured so re-running with a constant-t input would expose that asymmetry. --- ## Slide 8 — What the next builder should measure This is what closes the L3 → L4 gap. Build any one row, then capture: 1. **Strike-tone tap test on the unbored blank**, then on the finished tube. Compare to the structural FEM modes. Acceptance: lowest measured tap mode within 10% of `mode 1` for the matching L:D row. 2. **Bore impedance measurement at the labium end.** Acceptance: the first three peaks at `n * f0` with f0 within 0.5% of the design value, after end-correction calibration. 3. **Played fundamental + first three overtones** with a tuner. Acceptance: each within 5 cents of the harmonic-series target, averaged over 10 s of steady playing at moderate dynamics. 4. **Wall flexibility check.** Block the foot end, measure pressure inside the bore at the labium end, derive the effective bulk compliance, compare to the predicted -0.05-cent pull. Acceptance: sign correct and magnitude within an order of magnitude. 5. **Timbre fingerprint.** Long-time-average spectrum across L:D rows to validate the bright-vs-dark intuition the wider-bore row is supposed to deliver. Subjective rating + spectral centroid; no formal acceptance threshold (this is timbre research, not tuning). 6. **Build-quality data for the empirical correction loop.** Every row built becomes a row in `validation.csv` with measured f0, the four structural modes, the timbre fingerprint, and the player rating. Three rows per L:D = 12 instruments closes the empirical gate. Items 1..4 are validator-clean L4 packet inputs. Items 5..6 are the research output that justifies running the L:D sweep at all. --- ## Slide 9 — Reproducibility ``` git checkout sprint/claude-alice-r3-capstone-fem cd capstone python3 scripts/run_capstone.py ``` This regenerates every artefact in the deck. Inputs are pure code + material constants in `capstone/scripts/fujara_design_math.py`. No external data files. SHA256s of the generated outputs are tracked in `capstone/REPRODUCIBILITY.md`. ### FEM substitution disclosure The handoff allowed Wolfram or FEniCS for the modal and acoustic solves. Neither was installed in the worktree's Python environment (`pip install fenics-dolfinx mph` requires conda + Comsol respectively and was out of scope for this PR). The substitutes are: - **modal**: scipy cubic-Hermite FEM, validated against the closed- form free-free Euler-Bernoulli result. - **acoustic**: scipy linear FEM on the 1-D Helmholtz operator, validated against the closed-form open-open pipe result. - **coupled**: closed-form Smith-Korpela wall-flexibility perturbation, whose order-of-magnitude prediction a future FEniCS coupled solve is expected to reproduce. Each script accepts the same `FujaraRow` dataclass, so swapping in a 3-D FEniCS solve is a one-function replacement (`compute_modes` or `solve_cavity_modes`).