What the Inverse Square Law Covers and Where It Stops

Why this matters

Six decibels per doubling of distance is the one piece of acoustics that everyone in the trades knows, and it is the one most often applied where it does not hold. A shop measures a roof array at 4 m, extrapolates to the property line three doublings away, promises a number, and misses by enough to lose the argument with the neighbour. The rule is not wrong. It is a spreading law derived under three specific conditions, and in a real building at least one of them usually fails. Each failure changes the answer in a different, nameable way, and knowing which one you are in is the difference between an extrapolation and a wish.

What the rule actually says

Energy from a point source spreads over the surface of an expanding sphere. Double the radius and that surface area quadruples, so the intensity at any point on it drops to a quarter. A quarter of the power is 10 log(0.25) = -6.02 dB, which is the whole derivation.

Written against the source's own power level, and holding the source power constant, the source small compared with r, and the field free of reflections:

Lp = Lw - 10 log(S), with S the area of the spreading surface in square metres.

Everything that follows is about which of those three held-constant conditions your job breaks.

Condition one: the source behaves as a point

Fails for extended sources, and the failure is graded rather than binary.

• A line source (a long duct run, a row of identical units on a common curb, a pipe, a road) radiates over the surface of an expanding cylinder rather than a sphere. Double the distance and the area only doubles, so the level drops 3 dB per doubling, not 6. This holds while you are within roughly the array length divided by pi. Past that distance the array starts to look compact and the slope transitions toward 6 dB per doubling.
• A large plane radiating surface (a wall or deck being driven by structure-borne energy) gives close to no drop at all while you are near it and small compared with it, because the spreading has nowhere to go.

The practical consequence: measure the slope, do not assume it. The slope tells you what kind of source you have, which is often more useful than the level did.

Condition two: the field is free

Fails wherever reflected energy reaches the receiver, which indoors means past the critical distance, where the room's reverberant contribution flattens the curve to nearly nothing. That is the room card's territory and the arithmetic lives there.

Outdoors it fails near hard vertical surfaces. A microphone 1 m from a masonry facade is receiving the direct wave and its reflection, and depending on frequency and geometry those can add. It also fails over hard ground for a source and receiver both close to the ground, where the ground reflection is nearly coincident with the direct path.

Condition three: the source power is held constant

This is the condition nobody writes down, and it is the one that quietly voids a measurement. The 6 dB figure is a statement about spreading, not about the machine. If the source's emitted power changes between your two positions, the slope you measure is a mixture of spreading and source variation and cannot be separated after the fact.

The realistic ways it happens: a unit stages up or down between readings; wind or ambient temperature changes the load; a compressor cycles; you took the far reading twenty minutes later and the plant is doing something else. Log the operating state with every position, and if it changed, the pair is void.

What changes the offset without changing the slope

Directivity shifts the constant, not the exponent. A source radiating into a full sphere spreads its energy over 4 pi r squared; the same source on a large hard reflecting plane radiates into a half space, over 2 pi r squared, which is exactly 3.0 dB higher at every distance. In a floor-wall junction the factor is 4 and in a corner 8. All of these move the whole curve up or down. None of them changes the 6 dB per doubling slope, and anyone who quotes a modified slope because a unit sits on a roof has confused the two.

Worked example: the roof array that would not fall off at 6

A row of eight identical exhaust fans on a common curb, spanning 20 m along the roof edge. Individual fan largest dimension 0.6 m. A complaint from a property line about 32 m away.

Basis, held throughout: A-weighted, overall, 5-minute equivalent-continuous sound pressure level (LAeq,5min) re 20 micropascals, microphone on a stand at 1.5 m above the roof, on a line perpendicular to the array at its mid-length, all eight fans running at the same staging for both readings. Distances in metres throughout.

• At 2.0 m: 72.4 dB
• At 4.0 m: 68.8 dB

Measured slope: 3.6 dB per doubling.

Compare against both models before choosing one. The point-source model predicts 6.0 dB per doubling. The line-source model predicts 3.0 dB per doubling. The measured 3.6 is 2.4 below the point prediction and 0.6 above the line prediction, so this array is behaving as a line, not as a point, at these distances.

Check that against the geometry rather than accepting it. The line-source regime holds within roughly the array length divided by pi, which is 20 / pi = 6.4 m. Both measuring positions are inside that, so line behaviour is what the geometry predicts and the measurement agrees. The 0.6 dB above the pure line slope is consistent with 4.0 m being partway toward the transition.

Extrapolate to the property line, both ways.

The naive point-source extrapolation from 4.0 m: three doublings to 32 m, so 20 log(32/4) = 18.1 dB of spreading loss. 68.8 minus 18.1 = 50.7 dB.

The two-regime extrapolation: line behaviour from 4.0 m out to the 6.4 m transition, then point behaviour beyond it.

• 4.0 m to 6.4 m at the line slope: 10 log(6.4 / 4.0) = 2.0 dB
• 6.4 m to 32 m at the point slope: 20 log(32 / 6.4) = 14.0 dB
• Total spreading loss: 16.0 dB. 68.8 minus 16.0 = 52.8 dB

The gap is 2.1 dB, and it runs in the flattering direction, which is the direction that loses arguments. The naive model promises the neighbour a quieter property line than the geometry supports. Where an error costs somebody something, the rounding and the model choice both go the conservative way, so the figure that goes in the report is 52.8 dB with the two-regime derivation printed beside it.

Sibling-rule checks on this scenario, printed.

• Near field. The element being characterised at 2.0 m is an individual 0.6 m fan, and 2.0 m is 3.3 times that dimension, which clears the geometric near-field floor the sibling card sets. The array as a whole cannot be measured in its own far field at these distances, which is not a defect: it is exactly why the line model is the correct treatment rather than the point model.
• Constant source power. Both readings were taken at the same staging, minutes apart, and the staging was logged. The condition is satisfied and stated rather than assumed.
• Free field. Both positions are on an open roof away from the parapet, so no strong reflecting surface is within the measurement geometry. Had they been within a metre of the parapet, the pair would be void.
• Basis. One weighting, one bandwidth, one time basis, one height, one unit of distance, across both readings and both extrapolations.
• No new numbers. Every value in the extrapolation comes from the general section: the 6 dB point slope, the 3 dB line slope, the array-length-over-pi transition. Nothing was introduced to make the example resolve.

The failure mode. A shop that quotes 50.7 dB and installs a treatment sized for a 50.7 dB starting point will find the line 2 dB higher than the arithmetic said, which is a real complaint on a limit written to a whole decibel. The tell that the model was wrong was available for the cost of one extra reading: the measured slope was 3.6, not 6.0, and nobody had to know any theory to notice that.

Additions versus re-basings

Two things get applied to a propagation calculation and they are not the same operation.

Changing from full-space to half-space spreading is a re-basing. The same spreading term is being restated against a different radiating surface area. You replace 4 pi r squared with 2 pi r squared. You do not add 3 dB on top of a figure that already assumed half space, which is the double-count that shows up when a published pressure figure already contained a reflecting plane.

Atmospheric absorption and ground effect are additions. They are separate physical mechanisms that remove energy in addition to spreading, they are strongly frequency-dependent, and atmospheric absorption depends on temperature and humidity. At 32 m they are small; at several hundred metres they are not. They belong in a long-range calculation and the values belong to a propagation standard, in the edition your specification or ordinance names, not to a rule of thumb.

Checking a slope before you extrapolate

Take three positions, not two. Two points define a slope whether or not the slope means anything. Three positions at doubling distances tell you whether the slope is constant, which is what an extrapolation assumes.

Compare the measured slope to both candidate models before you pick one, and write down which you picked and why. A slope between 3 and 6 usually means you are near a transition, and the conservative choice is the model that predicts the higher level at the receiver.

Confirm the source did not move between readings, including its operating state, staging and load. This is the condition that leaves no evidence when it fails.

Roof work to take any of these readings is fall-exposed and the trigger height depends on the Part: 4 feet under 29 CFR 1910.28(b)(1) in general industry, 6 feet under 29 CFR 1926.501 in construction, so establish which covers the job before anyone steps out, and stay back from an unguarded edge while walking a microphone stand out along a line. Place the stand outside the plane of any rotating component with guards in place per 29 CFR 1910.212(a)(1), and never reach past a guard to position it.

References

• 29 CFR 1910.28(b)(1) for general industry fall protection and 29 CFR 1926.501 for construction, whichever Part covers the work
• 29 CFR 1910.212(a)(1) for machine guarding around a microphone position
• An outdoor sound propagation standard, in the edition your specification or ordinance names, for atmospheric absorption and ground effect at long range
• See related: Why the Same Machine Reads Differently in Two Rooms; What Near Field and Far Field Mean for a Measurement