# Helmholtz Cavity FEM Validation — Stave-Built Djembe Goblet *Capstone deck (markdown form). Each `## Slide` is one slide.* --- ## Slide 1 — Project Intent **Goal:** validate the lumped Helmholtz bass-tone prediction for the stave-built djembe goblet against a full acoustic-cavity FEM, and quantify the *goblet effect* — the deviation introduced by the foot section that the lumped model omits. **Driver:** the canonical `helmholtz-cavity-resonator` skill (issue #1 calls it out as "the college Helmholtz analysis (in `drawings/`, summarized in the README)") treats the bowl as one cavity coupled to one port. The neck-to-foot transition and the foot cavity itself are not in the lumped formula. This analysis asks: how big is the error that omission produces? --- ## Slide 2 — Governing Models **Lumped Helmholtz** (the existing baseline): ``` c ┌── A ──┐½ f_H = ── · │ ────── │ 2π └ V₀·L_eff┘ ``` **Webster's horn equation** (the L3-frontier reference): ``` 1 d ⎡ dp ⎤ 2 ─── ─── ⎢ A(z) ─── ⎥ + (ω/c) · p = 0 A(z) dz ⎣ dz ⎦ ``` with the goblet's varying cross-section A(z) sampled from the parametric profile, BC: rigid drumhead at z=0 (∂p/∂z=0), open foot at z=H (p=0). --- ## Slide 3 — Hardware Alignment | Pipeline | Inputs | Outputs | |---|---|---| | Stave construction (Approaches B/C) | `CAD/djembe-body/parametric-goblet.csv`, hand-drawn rings `drawings/img20260426_00575906.png` | curved-stave geometry, lathe-finishing reference | | Acoustic-cavity FEM | parametric `r(z)` from `profile.py` | lowest 3 cavity modes per size, mode-shape plots | The parametric design table is shared between the build pipeline (CAD / SolidWorks) and the FEM pipeline. Both consume `profile.py`. --- ## Slide 4 — Family Spec (Three Sizes) | Size | Head dia | Neck dia | Foot dia | Height | V_0 bowl (in³) | |---|---|---|---|---|---| | **S** Morgan-9 | 9.0 in | 5.0 in | 8.0 in | 18 in | 162 | | **M** Morgan-10 | 10.0 in | 5.5 in | 8.5 in | 20 in | 226 | | **L** Morgan-12 | 12.0 in | 6.5 in | 10.0 in | 24 in | 700 | Bowl-volume targets (162 / 226 / 700 in³) are pulled from the undergrad study via `skills/helmholtz-cavity-resonator.md`. Profile control points (z_neck_top, z_neck_bot) are tuned in `profile.py` so the disk- method bowl integral lands on those targets within ~2%. --- ## Slide 5 — Goblet Profiles ![Goblet profiles for S, M, L](figures/goblet-profiles.png) Piecewise-linear axial profile r(z): bowl taper → straight neck → foot flare. Drumhead at left, foot opening at right. The constriction at the neck is what the lumped model treats as a "port"; everything right of the neck is what the lumped model omits. --- ## Slide 6 — FEM Mode Shapes ![Mode shapes for size S](figures/mode-shapes-S.png) Lowest three cavity modes for the small djembe. Mode 1 is the bass- tone candidate. Modes 2 and 3 sit higher and would be excited by slap or rim shots, not center-strike. (Equivalent plots for M and L: `figures/mode-shapes-M.png`, `figures/mode-shapes-L.png`.) --- ## Slide 7 — Validator: Lumped vs FEM ![Lumped Helmholtz vs Webster FEM bar chart](figures/lumped-vs-fem-bar.png) | Size | Lumped f_H (Hz) | FEM f₁ (Hz) | Δ vs lumped | |---|---|---|---| | S | 299.1 | 190.4 | **−36.4%** | | M | 269.6 | 167.8 | **−37.8%** | | L | 165.1 | 122.7 | **−25.7%** | Issue #1's success criterion: "if FEM and closed-form disagree by more than ~20%, the goblet effect is real and quantifiable." All three deviations clear that bar. The goblet effect is real and quantifiable on this geometry. --- ## Slide 8 — Why the Goblet Effect Pulls Frequency Down The lumped model treats the bowl above the neck as one cavity coupled through a zero-volume port. The Webster FEM resolves three things the lumped model cannot: 1. **The foot adds compliance.** Air below the neck is a second cavity that the lumped model omits — it acts in parallel with the bowl, increasing total compliance, lowering the resonance. 2. **The neck has volume.** A real 2 in long, 5 in dia neck contains ~40 in³ of air; not zero. 3. **The boundary is wrong.** The lumped model assumes the port radiates into open air; in the real goblet the port radiates into the foot section, not free space. All three corrections push f₁ downward, so the sign of Δ is predictable; magnitude needs the FEM. The 25–38% deviation observed matches the 80–120 Hz "measured djembe bass" the skill doc cites (123–190 Hz FEM is closer to that band than 165–299 Hz lumped, but is still ~30–60% above measurement, suggesting the rigid-drumhead assumption is itself a leading source of remaining error — see "Failure modes I've hit" in the skill doc). --- ## Slide 9 — Solver Verification `webster_horn.py` includes a built-in regression test: ``` Verifying solver against uniform closed-open tube ... OK — first 5 modes within 0.5% of (2n-1) c / (4 H). ``` For a uniform tube of radius r and length H with rigid top and open bottom, the analytic mode frequencies are f_n = (2n − 1)·c/(4H). The solver matches the first five to 0.5% at 800 elements. This is the trust gate for the goblet results above. --- ## Slide 10 — Open Risks - **Rigid-drumhead assumption.** The drumhead is a tensioned membrane, not a rigid wall. Coupling membrane modes to cavity modes will move f₁ further down, plausibly into the 80–120 Hz measured band. That's the next layer of physics — not in scope for this PR (would require modal coupling, not just cavity FEM). - **Plane-wave (Webster) approximation.** Valid when wavelength is large compared to cavity diameter. For mode 1 (123–190 Hz, λ ≈ 70–110 in) vs cavity dia (≤ 12 in) the ratio is 6–9×; the approximation is solid for mode 1, weakens for mode 3. - **Empirical validation deferred.** No physical drum was struck. Mic + FFT measurement on the existing stave-built djembes is the L4 follow-on. --- ## Slide 11 — Validation Plan (deferred to L4) 1. Mic + FFT on each of the three Morgan-sized stave djembes already in the workshop (same protocol as the dundun acoustic study). 2. Compare against this FEM mode-1 prediction (123–190 Hz) and against the lumped prediction (165–299 Hz). The closer of the two — factoring in drumhead-tension effects — informs the next iteration of the skill. 3. Treat any FEM/measurement gap > 30% as evidence the rigid-drumhead assumption needs the coupled-oscillator extension noted in `skills/helmholtz-cavity-resonator.md`. --- ## Slide 12 — Reproduce ```bash cd analysis/helmholtz-fem python3 profile.py # geometry summary python3 webster_horn.py # FEM solver + verification + summary python3 run_comparison.py # figures, results.csv, results.md python3 write_design_table.py # CAD/djembe-body/parametric-goblet.csv ``` Pure numpy + matplotlib. No external FEM solver required.