What the Mass Law Predicts and Where It Fails

Why this matters

"Add mass" is the most repeated piece of noise-control advice in the trades, and it is right often enough that nobody checks when it stops being right. Mass law is a prediction for one specific thing: a limp, single-leaf, large panel in the frequency region between its own resonances and the onset of coincidence. Outside that region it does not under-predict slightly, it stops describing the panel at all, and it stops in a direction that costs money. Doubling the weight of the wrong wall can buy nothing, and on a stiff heavy element it can move a weakness down into the octave you cared about. All four conditions are checkable before anything is ordered.

The relationship, and what it holds constant

For a limp single panel in the mass-controlled region, transmission loss rises about 6 dB per doubling of surface mass, at constant frequency, and about 6 dB per doubling of frequency, at constant surface mass. Surface mass is mass per unit area, density multiplied by thickness. It is the only material property in the relationship: at constant surface mass, steel, lead, concrete and a limp sheet all predict the same thing, and any difference you see between them is one of the failure conditions below.

The widely used field-incidence approximation is transmission loss equal to twenty times the logarithm of the surface mass in kg/m2 multiplied by the frequency in Hz, minus 47. That constant carries its own condition: it approximates a limp single panel excited over the limited range of incidence angles a laboratory produces, in the mass-controlled region only. Measured data in that region commonly comes in nearer 5 dB per doubling of frequency rather than 6, so treat the mass-law figure as an upper bound on what a real panel will give you, written with one inequality and not as an interval, and take the working curve from the assembly's own tested report.

Two unit conversions you will need constantly: 1 lb/sq ft is 4.88 kg/m2, and the surface mass of a steel sheet is its density of about 490 lb/cu ft multiplied by its thickness in feet.

The gate: four conditions, checked before you order

Apply all four. If any one fails, mass law is not the tool and the answer has to come from a tested report for that construction.

1. Single leaf. One continuous panel, not two panels with a cavity. A studded partition with board on both faces is a double leaf and obeys different rules, which a sibling card owns.
2. Limp at your frequency. Your frequency of interest sits at least an octave below the panel's critical frequency. Above that, bending stiffness takes over and the panel develops a dip rather than a slope. The critical frequency falls as the panel gets thicker, so this condition gets harder to satisfy on exactly the heavy elements people reach for.
3. Above the panel's fundamental resonance. Below its lowest resonance a panel is stiffness-controlled, and its behaviour there is a function of how it is framed and supported rather than what it weighs.
4. Sealed. Mass law describes a surface, not an assembly. It has nothing at all to say about a gap, and the composite arithmetic that does is owned by the transmission-loss card.

Outcome one: the enclosure panel

A partial enclosure around a shop compressor, six panels totalling about 120 sq ft, currently 18 gauge sheet steel at 0.0478 in. Surface mass is 490 multiplied by 0.0478 divided by 12, which is 1.95 lb/sq ft, or 9.52 kg/m2. The question on the table is whether to rebuild it in 14 gauge at 0.0747 in, which is 3.05 lb/sq ft or 14.89 kg/m2.

Run the gate at 500 Hz. Single leaf: yes. Limp at 500 Hz: yes, since sheet steel this thin has a critical frequency up in the kilohertz, comfortably more than an octave above. Above the fundamental resonance: yes for a stiffened panel of this size. Sealed: no, and this is the condition the job will turn on. There is an unsealed cable penetration of about 1 sq ft.

Mass law first, ignoring the hole. At 9.52 kg/m2 and 500 Hz: twenty times the logarithm of 4761, minus 47, gives 26.6 dB. At 14.89 kg/m2: twenty times the logarithm of 7446, minus 47, gives 30.4 dB. The predicted gain is 3.9 dB, which is exactly 6 dB per doubling applied to a mass ratio of 1.56.

Now apply condition 4, using the composite method. Convert each transmission loss to a coefficient, area-weight, convert back.

The rebuild in heavier steel, with the hole left open, buys 0.5 dB. Sealing that one square foot and leaving the existing gauge alone buys 6.8 dB. The 3.9 dB that mass law promised was real and was never available, because condition 4 failed and the hole was carrying most of the transmitted power.

The order is therefore: seal, then re-measure, then decide about gauge. If the enclosure is rebuilt heavier, two things travel with that decision. Cutting and grinding coated sheet steel puts metal and coating particulate into the air, so cut with local exhaust or outdoors, wear eye and face protection against the chips, and where the coating is unknown treat the fume and dust as a respiratory hazard requiring a respirator selected and fit-tested under a program meeting 29 CFR 1910.134 rather than a nuisance dust mask. And an enclosure is a thermal decision as well as an acoustic one: enclosing a compressor or a motor restricts the cooling air it was rated with, so confirm the ventilation openings the equipment manufacturer requires and keep them, because a quiet machine that overheats is a shorter conversation than a loud one.

Outcome two: the concrete wall

Same gate, an opposite answer. A plant office is separated from the shop by a 4 in normal-weight concrete wall, and the proposal is to build it out to 8 in for a low-frequency complaint centred on the 250 Hz octave band.

Surface mass at 145 lb/cu ft: 4 in gives 48.3 lb/sq ft, which is 236 kg/m2, and 8 in gives 96.7 lb/sq ft, which is 472 kg/m2. An exact doubling, so mass law would predict 6 dB.

Run the gate at 250 Hz. Single leaf: yes. Above the fundamental: yes. Sealed: assume yes for the moment. Condition 2 is where it stops. A concrete slab is stiff and heavy, and its critical frequency lands low: for a 4 in normal-weight slab it sits somewhere in the region of a couple of hundred hertz, and it falls roughly in proportion to thickness, so the 8 in wall's lands about an octave lower again. At 250 Hz you are at or above coincidence on both walls. Condition 2 fails, so the 6 dB prediction does not apply and neither does the arithmetic that produced it.

Worse, the direction is against you. Doubling the thickness moves the critical frequency down by about an octave, which drags the coincidence weakness deeper into the speech range rather than out of it. The sibling card on the coincidence dip owns why that happens and how to locate it for a given material and thickness. What matters here is that the intuitive move, more concrete, is the one that puts the dip closer to the complaint.

The honest answer for this wall is that the mass-law calculation cannot be used, and that a real prediction needs the tested transmission loss for the specific construction, band by band, from a report naming its test method and edition. If the building is going to be modified anyway, a decoupled second leaf will outperform more concrete at this frequency for far less weight, subject to the conditions the double-wall card sets. And any penetration made through a rated wall in the process has to be closed with a firestop system tested for that assembly, under the building code in the edition your authority having jurisdiction has adopted and amended, that authority being a named role with power to interpret and enforce rather than whichever inspector arrives.

Sibling-rule check. The 6 dB per doubling figure is stated twice, once with frequency held constant and once with surface mass held constant, and it is only applied where all four gate conditions hold. The field-incidence constant of 47 is stated with the condition it was derived under and flagged as an upper bound with a single inequality. The composite arithmetic area-weights coefficients rather than averaging decibels, matching the transmission-loss card. The critical-frequency direction, falling as thickness rises, matches the coincidence card. No number appears in either outcome that the general section did not already state.

How to verify you got this right

Check the gate in writing before you check the arithmetic, because a correct calculation on a failed condition is the expensive mistake here. Write down which of the four conditions you confirmed and how. "Single leaf" confirmed by looking at a section drawing is worth something; assumed from the room side is not, since a furred-out face turns a single leaf into a double one without changing the room's appearance.

Afterwards, measure band by band and compare the shape against the prediction, not just the value. A mass-controlled panel should show transmission loss rising steadily with frequency across the mass-controlled region. A curve that rises and then dips has met its critical frequency, and the bottom of that dip tells you where it is. A curve that is flat and low across the whole range is not describing the panel at all, it is describing a leak or a flanking path, and adding mass to it will produce a second flat low curve at the same place.

References

• ASTM E90, laboratory measurement of airborne sound transmission loss, in the edition stated on the report for the construction you are predicting; ISO 10140 is the international counterpart
• 29 CFR 1910.134, respiratory protection, for the program, selection and fit-testing required where cutting or grinding coated steel generates airborne particulate and fume
• Equipment manufacturer documentation for required cooling-air openings before any machine is enclosed
• The building code as adopted and amended by your authority having jurisdiction, for firestopping any penetration made through a rated wall
• See related: What Transmission Loss Actually Measures; What the Coincidence Dip Is and Why It Lands Where It Does; Why a Double Wall Beats a Heavy Wall and When It Does Not