Why a Single Number Hides the Fault

Why this matters

A customer, a landlord or a code officer hands you one number and asks you to fix it. That number is a sum, and sums in acoustics are logarithmic, so the total belongs almost entirely to whatever is loudest. Silence a source sitting 10 dB below the dominant one and the total falls about 0.4 dB, which no field meter resolves and no complainant notices. Shops burn real hours removing the source that is easiest to reach and then have to explain why nothing changed. The number did not lie. It was never capable of showing that fault.

A level is five things, and a number carrying fewer is not a measurement

Before any arithmetic, a level has to declare five conventions. Drop one and two people compare numbers that were never comparable.

• Quantity. Sound power level (Lw, dB re 1 pW) is a property of the machine and does not change with where you stand. Sound pressure level (Lp, dB re 20 uPa) is a property of a position, and it falls with distance and rises with room reverberation. A manufacturer submittal quoting Lw and a complaint quoting Lp are not the same quantity, and the path between them is the whole job.
• Reference. 20 micropascals for pressure, 1 picowatt for power. A bare "dB" with no reference is a ratio, not a level.
• Weighting. A-weighting rolls off low frequency to approximate the ear's reduced sensitivity there at moderate levels. C-weighting is close to flat from roughly 32 Hz up and is used for peaks and for screening low-frequency content. Z is unweighted. The same fan can read 78 dB(A) and 90 dB(C) at one position, and both are correct.
• Bandwidth. Overall, or per band with the band center frequency named. An overall level and a 63 Hz octave-band level are different measurements of the same air.
• Time basis. F (125 ms) and S (1 s) are exponential time weightings for a fluctuating display. Leq is the equivalent-continuous level over a stated period, and the period is part of the number. Lpeak is a true instantaneous peak and is never F or S. Occupational measurements under 29 CFR 1910.95 are taken with slow response, so a level you gathered on fast is not the number that standard is written around.

Write levels the long way in your notes: LAeq,10min = 63.9 dB re 20 uPa, overall. It costs six characters and it survives being read by somebody else six months later.

Levels add logarithmically

Two sources combine as a power sum, not an arithmetic one: the total is 10 times the base-10 logarithm of the sum of each level's antilog. In the field you never need the logarithm, only this table. It gives the number of decibels to ADD to the higher of two levels, given the gap between them, for two uncorrelated sources at the same position, same weighting, same bandwidth and same time basis.

Two consequences fall straight out. Doubling the NUMBER of identical, uncorrelated sources adds 3 dB, holding the per-source level constant: eight identical condenser fans are 9 dB above one, not eight times anything. And once a gap reaches about 10 dB, the quieter source has stopped participating.

Subtracting is a re-basing, and it has a validity gate

Background correction is where the discipline usually goes. Your meter reading with the plant running already CONTAINS the background, so removing it is a re-basing of that reading, not a separate allowance you add somewhere else. Measure the total with the source on, measure the background with it off at the same position and same time basis, then correct by the difference:

• Difference of 10 dB or more: the background contributes less than about 0.4 dB. Report the reading as-is and say the correction was not needed.
• Difference between 3 and 10 dB: correct. At 7.6 dB the correction is about 0.8 dB off the reading; at 4 dB it is about 2.2 dB.
• Difference under 3 dB: the measurement is not usable as a source level. Report the result as an upper bound with one inequality sign, never as a value with a plus-or-minus, because you know only that the source is at or below what you read.

Worked example: four sources, one property-line number

A complainant reports plant noise at a shared property line. Before anything runs, agree the shutdown sequence with the facility operator and do not shut down any unit serving a life-safety, refrigeration-critical or process-critical load - that instruction is yours and the process trip it can cause is yours too. Wear hearing protection whenever you stand inside the plant area with equipment running, since 29 CFR 1910.95(c) sets the hearing-conservation action level at an 8-hour time-weighted average of 85 dB(A). All levels below are LAeq,10min, dB re 20 uPa, overall band, at the same property-line position, class 1 meter, calibrated before and after the set.

Each source was run alone with the others off:

The pump line is a bound, so the total is a bound too, and it is written that way. Because the pump sits about 18 dB under the exhaust fan, the bound moves the total by under 0.1 dB and nothing downstream depends on it.

Sum them with the table, largest first. 62.1 and 58.0 is a gap of 4.1 dB, so add 1.5: 63.6. Against 51.8 the gap is 11.8 dB, so add 0.3: 63.9. Against the pump bound the gap is about 19.9 dB, so add under 0.1: at or below 63.9 dB(A), LAeq,10min, re 20 uPa, overall. That matches the measured all-running total of 63.9, which is the check that the individual runs were clean.

Now price the obvious job. The compressor casing is the accessible one, in a plant room, nothing on the roof. Remove it entirely and the total goes to 63.6 - a gain of 0.3 dB. The complainant will hear no change at all, and the meter cannot defend the work.

Take 10 dB off the exhaust fan instead, the dominant source, and the fan lands at 52.1. Re-sum: 58.0 and 52.1 is a gap of 5.9 dB, add 1.0, giving 59.0; against 51.8 the gap is 7.2 dB, add 0.8, giving 59.8; the pump adds under 0.2. Total at or below 60.0 dB(A). A 10 dB reduction on the loudest source bought 3.9 dB overall, because the condenser fans were waiting underneath it.

If the ordinance needs 10 dB overall, the arithmetic says every source above about 54 dB(A) has to come down together. Treat the exhaust fan and the condenser fans both to 52 and leave the rest: three contributors near 52 sum to 56.8, plus the pump bound gives at or below 57.2 - a 6.7 dB gain, still short. Reaching 53.9 needs all four sources near 47 to 48. That is the honest scope, and it is worth putting in front of the customer before the first control is bought rather than after the second one disappoints.

Sibling-rule check. Every level in this example carries quantity, weighting, reference, bandwidth and time basis. The background correction is printed as a re-basing of the reading it belongs to, with its gap. The one figure failing the 3 dB validity gate is written as a bound with a single inequality. The summation table is used only for uncorrelated sources at a common position and basis, which is the condition it was stated under.

What would change the arithmetic

Correlated sources. The table assumes the sources are uncorrelated. Two units driven by the same shaft, or two openings radiating the same tone, can be phase-related, and coherent addition of two equal levels is up to 6 dB rather than 3 dB. If you hear beating - a slow rise and fall as two near-identical tones drift past each other - stop using the table and treat the pair as one source.

A different weighting. All of the above holds band by band and weighting by weighting, but you cannot mix. An A-weighted overall total cannot be decomposed into bands after the fact, because the weighting was applied before the summing. If you will need band data, collect band data.

Near field. Levels measured close enough to a large source that the reading changes sharply with small movements are near-field readings, and they do not sum or extrapolate reliably. Back off until the reading changes smoothly with distance.

How to verify you got this right

Sum the individual source levels and compare against the all-running measurement at the same position. Agreement inside about 1 dB means your shutdowns were clean and nothing was left running. A measured total noticeably ABOVE your sum means a source you did not isolate; a total noticeably BELOW it means one of your "individual" runs had a second machine on.

Then re-read your own report for the failure this article is about. If any sentence credits a control with a reduction, check that the control acted on a source within about 6 dB of the loudest one. If it did not, the sentence is arithmetically incapable of being true, however good the control is.

References

• 29 CFR 1910.95, Occupational noise exposure, for the hearing-conservation action level of an 8-hour time-weighted average of 85 dB(A) and for the slow-response measurement basis
• ANSI/ASA S1.4, sound level meter specification, in the edition your specifier or ordinance names; class 1 instruments are what a contested measurement needs
• Your local noise ordinance, which is the instrument that sets a community limit, its measurement position and its time basis; there is no federal community noise limit
• See related: What Octave Bands Tell You That an Overall Level Cannot; Reading a Vibration or Sound Signature as Its Own Diagnostic Method