Architectural Acoustics & Auditorium Timber Baffles, Ingard-Fok Perforated Resonant Panel
Perforated Acoustic Panel Porosity & Peak Hz Formula
Calculate Square-Grid Perforation Open-Area Porosity (%), Ingard-Fok Hole-Interaction Effective Neck Length teff (mm), Resonant Peak Absorption Frequency f0 (Hz), and Wavelength (m).
Ignoring Ingard-Fok hole-to-hole radiation end correction over-predicts perforated wood auditorium wall resonant frequency by 12%!
Perforated timber auditorium wall panels and micro-perforated ceiling baffles (`Hole Diameter d` mm on `Square Pitch b` mm across `Panel Thickness t` mm backed by a `Mineral-Wool Air Cavity of Depth D` mm) act NOT as porous foam, but as a **Coupled Array of Distributed Helmholtz Resonators**! Square-grid drilling fixes **Open-Area Porosity `epsilon = (PI() * d^2) / (4 * b^2)`**; however, oscillating air plugs inside each hole radiate acoustic mass beyond both faces while interacting hydrodynamically with neighboring holes per **Ingard (`1953`) & Fok’s Interaction End-Correction `t_{eff} = t + 0.8 * d * (1 - 1.47 * SQRT(epsilon) + 0.47 * epsilon^1.5)`** (`mm`)! Coupling air-plug inertance `t_{eff}` against cavity compliance `D` tunes **Peak Sound Absorption Frequency `f_0 = (c / (2 * PI())) * SQRT(epsilon / ((t_{eff} / 1,000) * (D / 1,000)))`** (`Hz`, at acoustic wavelength `lambda_0 = c / f_0` m) right onto `250–1,000` Hz human speech muddiness!
Verify on a `12.0` mm Birch Plywood Auditorium Acoustic Panel—Hole_Diameter `d` = 6.0 mm (B2), Square_Grid_Pitch `b` = 24.0 mm (B3), Panel_Thickness `t` = 12.0 mm (B4), Backing_Cavity_Depth `D` = 65.0 mm (B5), and `20°C` Speed_of_Sound `c` = 343.0 m/s (B6): Open-Area Porosity equals `4.91%` (E2), Ingard-Fok Effective Neck Length `t_{eff}` equals `15.26` mm (`+27%` longer than `12` mm wood alone! E3), Peak Absorption Frequency `f_0` tunes cleanly to **`384.0` Hz** (E4), and Resonant Wavelength equals `0.893` m (E5)!
| Ingard-Fok Perforated Absorber KPI (Panel t=12 mm, Sound Speed c=343 m/s) | Formula | Verified Output |
|---|---|---|
| Open-Area Porosity epsilon (%) | ((PI() * d_mm^2) / (4 * Pitch_b_mm^2)) * 100 | 4.91% low-mid tune (19.63% transparent) |
| Effective Neck Length teff (mm) | t + 0.8*d*(1 - 1.47*SQRT(eps) + 0.47*eps^1.5) | 15.26 mm (14.49 mm @ dense pitch) |
| Resonant Peak Frequency f0 (Hz) | (c / (2*PI())) * SQRT(eps / ((teff/1000)*(D/1000))) | 384.0 Hz warm speech (1,074.0 Hz @ 19.6%) |
| Resonant Acoustic Wavelength (m) | Speed_Of_Sound_c_m_s / Peak_Freq_f0_Hz | 0.893 m (0.319 m @ 1,074 Hz) |
Ingard-Fok Coupled-Aperture End Correction & Distributed Helmholtz f0 Formula
Validate `Hole_Diameter d` in `[0.3, 30]` mm, `Pitch b >= 1.5 * d`, `Panel_Thickness t` in `[0.5, 50]` mm, `Cavity_Depth D` in `[10, 600]` mm, and `Sound_Speed c` in `[310, 370]` m/s!
=LET(dm_h, B2, bp_m, B3, tp_m, B4, dc_m, B5, cs_m, B6, ep_s, (PI() * (dm_h ^ 2)) / (4 * (bp_m ^ 2)), te_f, tp_m + 0.8 * dm_h * (1 - 1.47 * SQRT(ep_s) + 0.47 * (ep_s ^ 1.5)), f0_h, (cs_m / (2 * PI())) * SQRT(ep_s / ((te_f / 1000) * (dc_m / 1000))), IF(OR(dm_h < 0.3, dm_h > 30, bp_m < 1.5 * dm_h, tp_m < 0.5, tp_m > 50, dc_m < 10, dc_m > 600, cs_m < 310, cs_m > 370), "Invalid Panel Input", ROUND(cs_m / f0_h, 3)))High-Porosity 19.6% Grille + Shallow 35 mm Furring (`d = 8` mm, `b = 16` mm, `D = 35` mm) Check
Drill `8.0` mm holes on tight `16.0` mm centers over shallow `35.0` mm furring (`B2 = 8`, `B3 = 16`, `B5 = 35`): Porosity E2 quadruples to `19.63%`, mutual hole interaction shrinks end correction (`t_{eff} = 14.49` mm in E3), and Resonant Peak E4 shifts nearly `3x` upward to **`1,074.0` Hz** (`0.319` m wavelength in E5)—leaving `250–500` Hz room boom completely untreated!
=IF(ISTEXT(E5), E5, LET(ep_s, (PI() * (B2 ^ 2)) / (4 * (B3 ^ 2)), te_f, B4 + 0.8 * B2 * (1 - 1.47 * SQRT(ep_s) + 0.47 * (ep_s ^ 1.5)), ROUND((B6 / (2 * PI())) * SQRT(ep_s / ((te_f / 1000) * (B5 / 1000))), 1)))Try the example
Use the original values or change them to test your understanding.