Why Decibels Do Not Add the Way Numbers Do

Why this matters

A shop with six identical rooftop condensers gets a noise complaint and promises the neighbour it will shut half of them down at night. Half the machines, half the noise: that is the sentence everyone in the room accepts. It buys 3.0 dB, which most listeners describe as barely different, and the complaint comes straight back with the shop's own promise attached to it. The arithmetic that would have caught this takes one line. Decibels are logarithms of ratios, so levels combine by energy, not by count, and the consequences run against intuition in a direction that reliably embarrasses whoever guessed.

A decibel is a ratio, not an amount

There is no such thing as a decibel of sound the way there is a gallon of water. A decibel is ten times the base-ten logarithm of a ratio between a quantity and a fixed reference. Change the reference and the same physical situation gets a different number, which is why the reference has to be written down every time: 20 micropascals for sound pressure, 1 picowatt for sound power.

The scale exists because the audible range of pressure spans roughly a million to one. Compressing that into a two-digit number is the whole point, and the price of the compression is that the numbers stop behaving arithmetically. Ten plus ten is not twenty in any sense that matters here.

Ten log or twenty log: the quantity decides

Which multiplier you use is set by what kind of quantity you have, not by preference.

• Power-like quantities (sound power, sound intensity, energy, mean-square pressure) use 10 log of the ratio to the reference.
• Amplitude-like quantities (sound pressure, and voltage in the electrical world) use 20 log of the ratio to the reference.

The two are consistent, not competing, because pressure squared is proportional to power. That gives the two anchor facts worth memorising, each stated with what is held constant:

• Doubling the acoustic power of a source, with distance and environment unchanged, raises the level by 3.01 dB.
• Doubling the pressure amplitude at a point, which is the same thing as quadrupling the power reaching that point, raises the level by 6.02 dB.

Anyone who quotes 6 dB for a doubling of machines has mixed the two.

Adding two levels: the difference is the only input

To combine two levels of the same quantity, at the same point, on the same weighting, bandwidth and time basis, from sources that are not phase-locked to each other, you do not need both numbers. You need their difference, and you add the amount below to the higher of the two.

More than two sources: combine the first pair, then combine that result with the third, and so on. Order does not matter.

The loudest source governs, and the gap is the whole story

Read the bottom of that table again. A source 10 dB below another contributes 0.4 dB to the total. A source 20 dB below contributes nothing you can measure.

This is the single most useful consequence of the log scale in the field, and it sets the order of work: fix the loudest contributor first, because until you do, nothing you do to the others can show up. A shop that quiets the second-loudest source by 10 dB and then cannot find the improvement on the meter has not made a measurement error. It made a spending error the table predicted.

The unit of analysis matters and it is per measurement position, per band. A source that is 12 dB down on the overall A-weighted level can still be the dominant contributor in one octave band, and if that band is the one the complaint is about, the overall figure has hidden the answer. That is a per-band question, and combining bands to an overall uses exactly the same table.

Halving the count always buys 3 dB, whatever the count was

For N equal, uncorrelated sources at the same point, the total sits 10 log N above one of them alone. So going from 8 to 4 is 10 log 8 minus 10 log 4, which is 3.0 dB. Going from 6 to 3 is 3.0 dB. Going from 2 to 1 is 3.0 dB. The count you started with does not enter.

That is the rule that kills the shut-half-of-them promise, and it also tells you what would work: level reduction here is a power question. Each 3 dB you want costs a halving of total emitted power, so 6 dB is a quartering and 10 dB is a reduction to one tenth.

Worked example: the bank that could not reach 6 dB

A property-line complaint against a bank of six identical condensers. The local ordinance, adopted and enforced by the authority having jurisdiction, requires a 6 dB reduction from the current condition at the property line, measured on the ordinance's own stated basis.

The measurement, on one stated basis throughout. All figures are A-weighted, overall, 15-minute equivalent-continuous sound pressure level (LAeq,15min) re 20 micropascals, at the ordinance's property-line position.

• All six running: 58.0 dB
• All six off, everything else in the neighbourhood unchanged: 46.0 dB

Background check, printed as its own line because the sibling rule requires it. The combined reading is 12.0 dB above the background. That is above the 10 dB margin at which the background correction falls under half a decibel, so no correction is applied and 58.0 stands as the source level. The correction that would have applied is 0.3 dB, and it is being declined on the stated rule, not ignored.

Back out one unit. Six equal sources sit 10 log 6 = 7.8 dB above one, so one unit alone contributes 58.0 minus 7.8 = 50.2 dB at that position. Check it forward: 50.2 plus 7.8 is 58.0. Closed.

Run the promise. Three units running: 10 log 3 = 4.8, so 50.2 plus 4.8 = 55.0 dB. The reduction delivered is 3.0 dB, exactly as the halving rule says, against 6.0 dB required. Half the machines delivers half the required reduction in decibels, which is a quarter of the required reduction in energy.

Find what the ordinance actually demands. The target is 58.0 minus 6.0 = 52.0 dB. Solving 50.2 plus 10 log N = 52.0 gives 10 log N = 1.8, so N = 1.5 units. There is no such thing as one and a half condensers, so count reduction cannot reach the target at all. Running one unit alone gives 50.2 dB, a 7.8 dB reduction, which clears the target but does not cool the building.

Carry the answer forward and check it is still measurable. Suppose the shop instead cuts each unit's emitted power by a factor of four, which is 6.0 dB per unit, and all six keep running: the bank lands at 52.0 dB. But the background is 46.0 dB, so the meter at that position will read 10 log(10 to the 5.2 plus 10 to the 4.6) = 53.0 dB, not 52.0. The difference between reading and background is now 7.0 dB, which is inside the range where the background correction is applied rather than declined, and the correction at 7.0 dB is 1.0 dB. Applying it: 53.0 minus 1.0 = 52.0 dB, which is the compliant figure. The demonstration only works if the correction is applied and stated, and the procedure for that belongs to the sibling card.

The failure mode. A shop that promised 50% quieter and delivered 3.0 dB has handed the complainant a written admission that the fix did not work. Worse, the second half of this example is where it turns costly: the same shop, having got the real fix in, would have read 53.0 against a 52.0 limit and concluded it had failed, when the uncorrected reading was never the source level in the first place.

Sibling-rule check on this scenario, printed. The 10 dB no-correction margin and the correction figures used here are the sibling card's rule and are applied here as stated, not restated with different values. The one basis (A-weighted, overall, 15-minute equivalent-continuous, re 20 micropascals, at the ordinance position) is held on every figure above. No level from a different position is combined with any level here. All six units are separate machines with independent drives, so they are treated as uncorrelated, which is the condition the addition table requires.

Where the addition rule stops

Correlated sources. The table assumes the sources are not phase-locked. Two sources emitting the same pure tone with a fixed phase relationship, or a tone and its own strong reflection off a nearby hard surface, can add to 6 dB above one of them or cancel to nearly nothing, depending on position. This is why a pure-tone complaint can vary hugely over a metre of listener movement while a broadband one barely changes.

Different bases. An A-weighted level and a C-weighted level of the same signal are not two sources and do not combine. Neither do a 1-second and a 15-minute figure, or a peak and an equivalent-continuous figure.

Peak values. Peak levels do not sum with this table at all. The 140 dB peak sound pressure level figure for impulsive noise in the footnote to Table G-16 at 29 CFR 1910.95(a) is a peak criterion and is evaluated as one, not by combining anything.

Any test that requires staging a shutdown, as this one does, is a process change and not just a measurement: do not stop units serving a life-safety, medical, process-cooling or temperature-critical load without the agreement of whoever owns that load, and confirm with the controls owner that the sequence will not short-cycle compressors or trip the remaining machines on head pressure when the bank is unbalanced. Where the work to get the readings puts anyone on a roof, name the Part first, because the fall protection trigger is 4 feet under 29 CFR 1910.28(b)(1) in general industry and 6 feet under 29 CFR 1926.501 in construction.

References

• 29 CFR 1910.95, Occupational noise exposure, including the 140 dB peak figure for impulsive noise in the footnote to Table G-16 at (a)
• The local noise ordinance adopted by your authority having jurisdiction, which owns the limit, the measurement position and the time basis
• 29 CFR 1910.28(b)(1) or 29 CFR 1926.501 for fall protection, depending on which Part covers the work
• See related: How to Add and Subtract Noise Levels Correctly; Why Sound Power and Sound Pressure Are Not the Same Number