Architectural & Noise-Control Acoustics, Sound Isolation & Partitions
Acoustic Mass Law Wall Transmission Loss & Coincidence Formula
Calculate Partition Surface Mass Density m_s (kg/m²), Field-Incidence Mass-Law Sound Transmission Loss TL (dB), Critical Bending-Wave Coincidence Frequency f_c (Hz), and STC Speech-Band Warning.
Doubling wall thickness adds +6 dB Mass-Law isolation—yet HALVES critical coincidence frequency f_c into the speech band!
Below bending stiffness resonance, airborne sound isolation across a non-porous single-leaf partition of density rho (kg/m³) and thickness h (m -> surface mass m_s = rho * h kg/m²) obeys the Diffuse Field-Incidence Acoustic Mass Law (speed of sound in air c_0 = 343 m/s): TL = 20 * LOG10(m_s * f) - 47.3 dB—gaining +6.02 dB every time surface mass OR frequency f (Hz) doubles! However, every solid panel also carries dispersive flexural bending waves whose speed rises with SQRT(f) until matching c_0 at the Critical Coincidence Frequency f_c = (c_0^2 / (2 * PI() * h)) * SQRT((12 * rho * (1 - nu^2)) / (E_GPa * 10^9)) Hz! At f_c, trace grazing sound waves excite severe resonant "coincidence dip" leakage—explaining why limp heavy materials (lead/mass-loaded vinyl, high rho, low E) outperform stiff thick panels whose f_c drops right into the 500–2,000 Hz human speech band!
Verify on a 150 mm Cast-Concrete Studio Wall—Material_Density rho = 2,400.0 kg/m³ (B2), Wall_Thickness h = 0.15 m (B3 -> Surface Mass m_s = 360.00 kg/m² in E2), Test_Frequency f = 500.0 Hz (B4), Youngs_Modulus E = 25.0 GPa (B5), and Poisson_Ratio nu = 0.20 (B6): Field-Incidence Mass-Law Transmission Loss hits 20 * LOG10(180,000) - 47.3 = 57.81 dB (E3), while Critical Coincidence Frequency lands down at 131.27 Hz (E4, safely below the 500–4,000 Hz intelligibility band -> "ABOVE COINCIDENCE DIP" in E5)!
| Acoustic Partition KPI (Concrete 2,400 kg/m³, h=0.15 m, f=500 Hz, E=25 GPa) | Formula | Verified Output |
|---|---|---|
| Surface Mass Density ms (kg/m²) | Density_kg_m3 * Thickness_m | 360.00 kg/m² (720.00 @ h = 0.30 m) |
| Field-Incidence Mass-Law TL (dB) | 20 * LOG10(ms * Freq_Hz) - 47.3 | 57.81 dB (63.83 dB @ h = 0.30 m) |
| Critical Coincidence Freq fc (Hz) | (343^2/(2*PI()*h))*SQRT(12*rho*(1-nu^2)/E) | 131.27 Hz (65.64 Hz @ h = 0.30 m) |
| 6 dB Mass-Doubling Gain Check | +20 * LOG10(2) = +6.02 dB | +6.02 dB |
Field-Incidence Mass-Law TL & Critical Bending Coincidence Formula
Validate Density > 50, Thickness h > 0.001 m, Frequency f >= 20 Hz, Youngs_Modulus E_GPa > 0.01, and Poisson_Ratio nu in [0, 0.499]!
=LET(rh_o, B2, th_m, B3, fr_q, B4, em_g, B5, nu_p, B6, IF(OR(rh_o <= 50, th_m <= 0.001, fr_q < 20, em_g <= 0.01, nu_p < 0, nu_p >= 0.5), "Invalid Wall Input", LET(ms_k, rh_o * th_m, ROUND(20 * LOG10(ms_k * fr_q) - 47.3, 2))))6 dB Mass-Doubling & Inverse Coincidence Shift Check (Double Thickness h = 0.30 m)
Double Wall_Thickness B3 from 0.15 m to 0.30 m: Surface Mass E2 doubles to 720.00 kg/m², adding +6.02 dB to Mass-Law TL E3 (rising from 57.81 dB to 63.83 dB) while halving Critical Coincidence Frequency E4 from 131.27 Hz down to 65.64 Hz!
=IF(ISTEXT(E3), E3, ROUND(((343 ^ 2) / (2 * PI() * B3)) * SQRT((12 * B2 * (1 - (B6 ^ 2))) / (B5 * 1000000000)), 2))Try the example
Use the original values or change them to test your understanding.