Formula
Mass law (field incidence)
\[ R = 20 \log_{10}(m_s\, f) - 48 \]
What the symbols mean
- R — sound reduction index (dB)
- mₛ — surface mass of the panel (kg/m²)
- f — frequency (Hz)
- f_c — critical (coincidence) frequency (Hz)
- η — loss factor / internal damping (dimensionless)
- E — Young's modulus of the material (Pa)
- h — panel thickness (m)
- ρ₀c₀ — characteristic impedance of air (≈ 415 rayl)
Inputs
Internal damping of the material (dimensionless). Drives the depth of the coincidence dip: low for steel/glass, high for laminated glass or lead.
Result
Sound reduction index R = f(frequency)
The dip on the curve is the critical (coincidence) frequency, where the mass law stops applying.
Understand the theory — physics concept and calculationsinteractive
The sound reduction index R (in dB) measures how well a wall blocks the sound passing through it: the higher R, the better the insulation. For a single panel a remarkably robust rule governs the insulation across a wide frequency range — the mass law. It says that almost everything hinges on a single quantity: the panel's surface mass \(m_s\) (in kg/m²). Double the mass, or double the frequency, and you gain about 6 dB. Drag the slider below to watch the transmission-loss line rise as the wall gets heavier.
Where the mass law comes from
When a sound wave hits a wall it sets it vibrating; the wall then radiates sound on the far side. If the wall is governed by its inertia (its mass) alone, the heavier it is, the more it resists being moved, and the less it re-radiates. At normal incidence the theory gives:
where \(m_s\) is the surface mass (kg/m²), \(f\) the frequency (Hz), \(\rho_0 c_0\) the air impedance (≈ 415 rayl). In practice sound arrives from all directions (diffuse field): one uses the field-incidence mass law, which subtracts about 5 dB and gives the formula used by the tool:
Each doubling of \(m_s\) adds \(20\log_{10} 2 \approx 6\) dB, and each octave (doubling of \(f\)) adds the same: the famous 6 dB/octave slope.
The critical frequency (coincidence)
A real wall is not just a mass: it also has bending stiffness. It can therefore carry bending waves whose wavelength varies with frequency. At the critical frequency \(f_c\), the bending wavelength equals that of the airborne wave at grazing incidence: coincidence occurs, the wall becomes "transparent" and R collapses. It is computed from Young's modulus \(E\), density \(\rho\), Poisson's ratio \(\nu\) and thickness \(h\):
Practical consequence: a heavy but flexible wall (thick concrete) has a low \(f_c\) — often in the bass — whereas a light, stiff sheet (glass, thin steel) has its \(f_c\) up in the treble, right inside the measurement band. That is why a plain pane of glass "whistles" at high frequencies.
Above f_c: the role of damping
In the coincidence region the insulation is no longer driven by mass alone but by the material's internal damping \(\eta\) (the loss factor). Cremer's model gives:
The larger \(\eta\), the shallower the dip and the faster R recovers above \(f_c\). That is the whole point of damped materials (laminated glass, sandwich panels, lead): they "fill in" the coincidence dip. Conversely, thin steel (\(\eta \approx 0.0005\)) shows a spectacular dip.
Above a so-called critical frequency \(f_c\), the panel's bending waves sync up with the incident sound wave: the panel "lets the energy through" and R drops sharply — the coincidence dip. Its depth depends on the material's damping \(\eta\). Adjust \(f_c\) and \(\eta\) to watch the dip move and deepen.
From R(f) to a single number: Rw and STC
The R(f) spectrum is precise but unwieldy for comparing products. It is summarised by a single-number weighted index. In Europe that is \(R_w\) (ISO 717-1): a reference curve is slid against the third-octave spectrum (100–3150 Hz) until the sum of unfavourable deviations reaches but does not exceed 32 dB; \(R_w\) is the value of the shifted curve at 500 Hz. In the US the equivalent is STC (ASTM E413), of very similar logic. Caution: the \(R_w\) the tool estimates rests on the theoretical model of a single homogeneous panel; a certified \(R_w\) requires a laboratory measurement (ISO 10140).
Limits of the model
The model assumes a single, homogeneous, infinite, freely mounted panel. It ignores: double walls (partitions with an air gap, double glazing) that far exceed the mass law thanks to the mass-spring-mass resonance; edge effects and finite size, which raise R at low frequencies; flanking paths and rigid links; and very thick walls (concrete, brick) where shear waves lower R in the treble. For those cases one uses fuller models (Sharp for double walls, finite elements, measurements). For a single panel, mass law + coincidence remains the go-to sizing tool.
Worked examples
| Wall | Index | Reading |
|---|---|---|
| Plasterboard (12.5 mm) | Rw ≈ 29 dB | fc ≈ 2.8 kHz, dip in the treble |
| Single glazing 4 mm | Rw ≈ 29 dB | light but fc ≈ 3 kHz, awkward coincidence |
| Concrete 200 mm | Rw ≈ 54 dB | huge mass, very low fc (~95 Hz) |
| Steel 1 mm (low η) | Rw ≈ 28 dB | deep coincidence dip near 12 kHz |
- Believing mass solves everything: above \(f_c\) it is damping that matters.
- Applying the simple mass law to a double wall: it badly underestimates the real insulation.
- Forgetting that \(f_c\) depends on thickness: doubling a panel's thickness lowers its critical frequency.
- Confusing \(R\) (intrinsic to the wall) with the field insulation \(D_{nT}\) between two rooms (which also depends on volume and reverberation).
- Taking the theoretical \(R_w\) for a certified value: only a lab measurement is authoritative.
Related tools
Sources : B. H. Sharp, Prediction Methods for the Sound Transmission of Building Elements, Noise Control Eng. (1978); L. Cremer, M. Heckl, E. Ungar, Structure-Borne Sound (Springer); F. Fahy, Sound and Structural Vibration; D. Bies & C. Hansen, Engineering Noise Control; ISO 717-1 (weighted index \(R_w\)); ISO 10140 (laboratory measurement).