The Mass Law in Sound Insulation

When a sound wave strikes a wall, part of its energy is reflected, part is absorbed, and part passes through to the other side. Acoustical engineers quantify a wall’s ability to impede this transmission using a quantity called Sound Transmission Loss (TL). The most fundamental principle governing TL is the Mass Law: heavier walls block sound more effectively, and higher frequencies are attenuated more than lower ones.

Defining Sound Transmission Loss

The sound power transmission coefficient $tau$ is the ratio of transmitted acoustic power to incident acoustic power at the wall surface:

$$tau = frac{W_{text{transmitted}}}{W_{text{incident}}}$$

Transmission Loss is then expressed in decibels:

$$TL = 10 cdot log_{10}!left(frac{1}{tau}right) quad [text{dB}]$$

A wall with $TL = 40,text{dB}$ reduces the transmitted sound power by a factor of 10,000. In practice, even a few decibels of improvement is acoustically significant.

Physical Derivation of the Mass Law

Treat the wall as a driven oscillator. An incident plane wave produces a fluctuating pressure $p_i , e^{jomega t}$ that drives the wall panel. For a wall with surface mass density $m$ (kg/m²), Newton’s second law per unit area gives:

$$jomega m , v_w = p_i – p_r – p_t$$

At frequencies well below the coincidence frequency, wall stiffness and damping are negligible compared to inertial forces (mass-controlled region). Applying the continuity of particle velocity at the air–wall interface and solving the coupled wave equations yields:

$$tau approx left(frac{2rho_{text{air}} c}{omega m}right)^2 = left(frac{rho_{text{air}} c}{pi m f}right)^2$$

Taking the logarithm and simplifying with standard air properties ($rho_{text{air}} approx 1.21,text{kg/m}^3$, $c approx 343,text{m/s}$) for field-incidence conditions (diffuse sound field):

$$boxed{TL approx 20log_{10}(m cdot f) – 48 quad [text{dB}]}$$

This is the Field Incidence Mass Law. It predicts a 6 dB increase in TL for every doubling of either surface mass or frequency.

Coincidence Frequency and the Coincidence Dip

The mass law does not hold across all frequencies. At a critical frequency called the coincidence frequency $f_c$, the bending wave velocity in the wall panel exactly matches the speed of sound in air. The wall then resonates efficiently with the incoming wave, and the transmission coefficient rises sharply — causing a pronounced drop in TL known as the coincidence dip:

$$f_c = frac{c^2}{1.8 , c_L , h}$$

For 6 mm glass ($c_L approx 5200,text{m/s}$), $f_c approx 2000,text{Hz}$ — squarely within the most sensitive range of human hearing. For 100 mm concrete ($c_L approx 3800,text{m/s}$), $f_c$ falls below 200 Hz where hearing sensitivity is lower, making concrete a far more effective insulator across the speech range.

Three Regions of the TL Curve

A plot of TL versus frequency reveals three distinct regions:

• Mass-controlled region ($f ll f_c$): TL rises at approximately 6 dB/octave (20 dB/decade), governed by the mass law.
• Coincidence dip (near $f_c$): TL drops sharply as bending resonance couples efficiently with airborne sound.
• Above coincidence ($f gg f_c$): Stiffness and internal damping dominate; TL still increases but at a reduced slope of approximately 10 dB/decade.

In practice, the coincidence dip can reduce TL by 10–15 dB relative to the mass-law prediction. Increasing the internal damping (e.g., using laminated glass with a viscoelastic interlayer) mitigates this dip.

Double Walls and Flanking Transmission

A powerful engineering solution is the double wall (mass–air–mass system): two independent panels separated by an air cavity. At mid and high frequencies, the total TL approaches the sum of the individual panel TL values — effectively doubling the insulation for only a modest increase in mass. The governing relation for the cavity resonance frequency is:

$$f_0 = frac{c}{2pi}sqrt{frac{2rho_{text{air}}}{m_{text{eff}} cdot d}}$$

where $d$ is the cavity depth and $m_{text{eff}}$ is the effective combined surface mass. Below $f_0$ the double wall performs no better than a single wall of the same total mass.

However, flanking transmission — sound energy bypassing the wall via structural connections (floor slabs, ceiling ties, pipe penetrations) or through unsealed gaps — can completely dominate and limit the achieved TL to values well below the theoretical prediction. A 1 mm unsealed gap in a 40 dB wall can reduce its effective TL to below 25 dB. Airtight sealing of all penetrations is therefore as important as the wall mass itself.

Practical Examples

• 100 mm concrete wall ($m approx 230,text{kg/m}^2$): $TL_{500,text{Hz}} approx 20log_{10}(230 times 500) – 48 approx 52,text{dB}$ — suitable for residential party walls.
• 6 mm float glass ($m approx 15,text{kg/m}^2$): $TL_{500,text{Hz}} approx 20log_{10}(15 times 500) – 48 approx 31,text{dB}$ — adequate for office glazing, with the coincidence dip near 2 kHz.
• 12.5 mm gypsum board ($m approx 10,text{kg/m}^2$): $TL_{500,text{Hz}} approx 27,text{dB}$ — typical interior partition; double-layer assemblies are standard in recording studios.

These examples illustrate why concrete dominates in high-demand acoustic environments, while lightweight assemblies require double-wall construction and careful sealing to compensate for their lower surface mass.

References

1. Fahy, F. & Gardonio, P. (2007). Sound and Structural Vibration: Radiation, Transmission and Response, 2nd ed. Academic Press, Oxford.
2. Beranek, L. L. & Vér, I. L. (2006). Noise and Vibration Control Engineering: Principles and Applications, 2nd ed. Wiley, Hoboken.
3. Cremer, L., Heckl, M., & Petersson, B. A. T. (2005). Structure-Borne Sound: Structural Vibrations and Sound Radiation at Audio Frequencies, 3rd ed. Springer, Berlin.
4. ISO 717-1:2013. Acoustics — Rating of Sound Insulation in Buildings and of Building Elements — Part 1: Airborne Sound Insulation. ISO, Geneva.