Why Sound Power and Sound Pressure Are Not the Same Number

Why this matters

Two vendors quote the same duty. One sheet says 88, the other says 68, and the second one wins the job. Then the unit goes in and the neighbour complains anyway. Nobody lied: those were two different quantities with two different references, and the smaller number was not the quieter machine. Sound power belongs to the source. Sound pressure belongs to a source in a particular place, at a particular point, with a particular room or roof around it. Only one of those two travels between jobs, and a shop that cannot tell them apart will lose an argument it should have won.

Two quantities, two references

Sound power level, written Lw, is the total acoustic energy the source emits per unit time, expressed in decibels relative to 1 picowatt. It is a property of the machine, the way rated capacity is. Move the machine to a different room and Lw does not change. You cannot measure it directly with a microphone; it is derived from a set of pressure or intensity measurements taken on a defined surface under a defined test method.

Sound pressure level, written Lp, is the fluctuating pressure at one point, expressed in decibels relative to 20 micropascals. It is what a microphone senses and what an ear senses. It is a property of the source and the distance and the surroundings, all three at once. Move the machine and it changes. Move the microphone and it changes.

The references are not interchangeable and neither are the decibels built on them. Both are decibels; neither is "the" decibel. A figure of 88 dB re 1 pW and a figure of 88 dB re 20 micropascals describe entirely different things about entirely different objects.

What a pressure figure cannot do

This is the useful half of the subject, because most field mistakes are a pressure number being asked to do work it structurally cannot.

It cannot be compared with a pressure figure from another site. Two readings taken at the same distance from two identical machines in two different rooms will differ, sometimes by 10 dB or more, and both are correct. The rooms differ, not the machines. A sibling card covers why the same machine reads differently in two rooms; the point here is that the comparison is void, not merely noisy.

It cannot be moved to a different distance without a geometry. Extrapolating a pressure figure to another distance requires knowing whether the source behaves as a point or a line, whether the field is free or reverberant, and what surface the source sits on. Those conditions and the way each one fails are their own card. Without them, moving the number is a guess wearing arithmetic.

It cannot be checked against a limit taken somewhere else. A community noise ordinance is a local instrument, adopted and enforced by the authority having jurisdiction, which is a named role with the power to interpret and enforce it and not a synonym for whoever shows up. The ordinance names its own measurement position, weighting and time basis. A reading taken 1 m from the unit is not evidence about a limit written at a property line, in either direction.

It cannot be de-rated for the room afterward. There is no correction you apply to a finished pressure reading to recover "what the machine really is". The room's contribution is baked in at the moment of measurement.

It cannot be summed with a pressure figure taken elsewhere. Levels add by energy only when they are levels of the same quantity at the same point on the same basis. Two machines' pressure levels measured at two different positions do not combine into anything.

What a pressure figure is exactly right for

Three jobs, and it is the correct and only quantity for all three.

Exposure. 29 CFR 1910.95 is written in sound pressure level at the employee, and 1910.95(d)(1) requires monitoring wherever information indicates any employee's exposure may equal or exceed an 8-hour time-weighted average of 85 dBA. A sound power level tells you nothing directly about anyone's exposure.

Complaint. A person hears pressure at their ear, in their room. When a tenant says it is too loud, the honest measurement is a pressure level at the position they occupy.

Before and after at one fixed position. The single most useful field measurement in this trade is the same position, same weighting, same time basis, before and after a change. The room contribution is identical in both readings, so it cancels out of the difference even though neither absolute number transfers anywhere.

The conversion, and what it already contains

In a free field, with the source small compared with the measuring distance, at standard atmospheric conditions and with no reverberant contribution, the pressure level at distance r relates to the power level by the area of the surface the energy is spread over:

Lp = Lw - 10 log(S), with S in square metres.

For a source radiating into a full sphere, S is 4 pi r squared. For a source sitting on a large hard reflecting plane, radiating into a half space, S is 2 pi r squared. Those two differ by exactly 3.0 dB at every distance, because the same energy is spread over half the area.

Here is the part that gets missed. When you convert a published pressure figure back to a power level, you are not adding a correction on top of the number. You are re-basing it: the published pressure already contains a distance and an environment, and you are removing that environment and restating the figure on the source's own basis. If you treat it as an addition you will apply the geometry twice. Say which one you are doing, in the same breath as the arithmetic.

Worked example: two spec sheets, one duty

Two condensing units are shortlisted for the same replacement.

Unit X publishes 88 dB re 1 pW, A-weighted, overall, determined under a named sound power test method over a reflecting plane, in the edition the sheet cites.

Unit Y publishes "68 dB at 10 feet". No weighting stated, no time basis, no environment.

Step 1, match the basis before anything else. Unit X's figure is a power level. Unit Y's is a pressure level. They cannot be compared as printed, and the fact that 68 is smaller than 88 carries no information at all.

Step 2, convert Unit Y's distance into the unit the geometry uses, and hold it there. 10 ft is 3.048 m. Every figure below is in metres.

Step 3, apply the re-basing, and print both branches because the sheet did not say which environment it assumed.

• Half space, on a reflecting plane: S = 2 pi (3.048 squared) = 58.4 square metres. 10 log 58.4 = 17.7. Lw = 68 + 17.7 = 85.7 dB re 1 pW.
• Full space, free field: S = 4 pi (3.048 squared) = 116.7 square metres. 10 log 116.7 = 20.7. Lw = 68 + 20.7 = 88.7 dB re 1 pW.

Step 4, print the qualifiers the general section imposed, each as its own line, because a qualifier that never appears in the arithmetic has not been applied.

• Weighting: Unit X is A-weighted. Unit Y states none. If Unit Y's 68 is A-weighted the comparison stands as computed; if it is unweighted, its A-weighted figure is lower by an amount set by its spectrum, so the comparison is not yet closed. Recorded as unresolved rather than assumed.
• Bandwidth: both figures are overall, not per band. Matched.
• Time basis: Unit X's method produces a steady-state figure. Unit Y states none. Recorded as unresolved.
• Geometry held constant: the conversion assumes a point source, a free field with no reverberant term and standard atmospheric conditions. A condensing unit at 3.048 m is not much more than one unit dimension away, so this conversion is an approximation, and it is being labelled as one rather than presented as a measurement.
• Correction character: this is a re-basing of Unit Y's published figure, not a term added to it. The 17.7 or 20.7 is not stacked on top of an environment the number already carried; it replaces it.

Step 5, read the result honestly. Unit Y is either 2.3 dB quieter than Unit X or 0.7 dB louder, and the entire 3.0 dB span between those two answers is the unstated half-space-versus-full-space assumption, not a property of Unit Y. The selection cannot be made from these two sheets.

Step 6, the failure mode. A buyer who reads 68 against 88 concludes Unit Y is 20 dB quieter, which would be a difference the neighbour could not miss. The real spread between the two candidates is somewhere under 3 dB, which is at the edge of what a listener notices at all. That buyer has paid for a quiet machine and will get a complaint, and the sheet that misled them contained no false statement.

Asking for the number that travels

The request that ends this problem is short, and it is worth having as a standing line in your submittal reviews: give me the A-weighted sound power level in dB re 1 pW, octave band and overall, with the test method and edition it was determined under. Every one of those clauses does work. Power because it transfers. A-weighted plus octave band because the overall alone will not let anyone choose a treatment. The method and edition because the mounting and the surface are part of the number.

If a vendor can only supply a pressure figure, the follow-up is equally short: at what distance, over what surface, with what weighting and what time basis, and was any background correction applied. A supplier who can answer those has given you something usable. One who cannot has given you a marketing figure, and the correct action is to record that in the submittal rather than to convert it and pretend.

Where you do go measure a candidate in the field: rooftop access is fall-exposed, and the trigger height differs by Part, 4 feet under 29 CFR 1910.28(b)(1) for general industry and 6 feet under 29 CFR 1926.501 for construction, so name which one covers the job before you go up. Set the microphone on a stand outside the plane of any rotating component with guards in place per 29 CFR 1910.212(a)(1), never reaching past a guard to place it, and wear hearing protection in any space where you must raise your voice to be heard at arm's length. Protection on your ears does not change what the microphone reads.

References

• 29 CFR 1910.95, Occupational noise exposure, including the monitoring trigger at (d)(1)
• ISO 3744 and the related sound power determination methods, which bind only through the purchase specification, ordinance or test report that names them, in the edition named there
• 29 CFR 1910.212(a)(1) for machine guarding, and 29 CFR 1910.28(b)(1) or 29 CFR 1926.501 for fall protection depending on the Part your work falls under
• The local noise ordinance adopted by your authority having jurisdiction, which owns any property-line limit
• See related: Why the Same Machine Reads Differently in Two Rooms; What the Inverse Square Law Covers and Where It Stops