Reference
Six decibels a doubling, and then the air takes its share
Two things separate the level at the front row from the level at the back. One is geometry: the same power spread over a bigger sphere. The other is the air itself, which turns a little of the sound into heat and does so far more readily at high frequencies than low. The first is the same at every frequency and is the one everybody quotes. The second is what actually makes the back of a field sound dull.
Below: the inverse square law and its consequences, why a line array only obeys it beyond a distance that depends on frequency, ISO 9613-1 air absorption computed in full rather than read off a chart, and a calculator that combines the two for your array, your distance and your weather.
The inverse square law
A source small compared with the distance to it radiates into a sphere. The area of that sphere grows with the square of the radius, so the intensity through any patch of it falls with the square of the distance — hence the name. Sound pressure goes as the square root of intensity, and level is twenty times the log of pressure, so the loss going from distance d₁ to d₂ is
ΔL = 20 · log₁₀ (d₂ / d₁)
which is 6.02 dB for every doubling and 20 dB for every tenfold. The figure has no frequency in it. Whatever the source, whatever the weather, the sphere is the same size, so this is the part of the loss that is identical at 50 Hz and at 10 kHz.
The loudspeaker convention of stating sensitivity at one metre for one watt (or 2.83 V, which is one watt into eight ohms) hangs off this. Level at any distance from a point source is then
L = S + 10 · log₁₀ (P) − 20 · log₁₀ (d)
with S the sensitivity, P the power in watts and d in metres. Ten times the power buys 10 dB; ten times the distance costs 20. The asymmetry is the whole economics of covering a large audience from one place, and the reason it is not done.
| Distance | Loss from 1 m |
|---|---|
| 1 m | −0.0 dB |
| 2 m | −6.0 dB |
| 4 m | −12.0 dB |
| 8 m | −18.1 dB |
| 16 m | −24.1 dB |
| 32 m | −30.1 dB |
| 64 m | −36.1 dB |
| 128 m | −42.1 dB |
A cabinet that makes 130 dB at one metre makes 88 dB at 128 metres, before the air has done anything. Every row is 6 dB off the last, and there is nothing to be done about it except start louder, start closer, or stop being a point source.
- It is a free-field law. Indoors, the direct sound obeys it and the reverberant field does not — past the critical distance the level stops falling because the room's contribution has caught up. Outdoors it holds until ground, wind and temperature gradients intervene, which they do over a few tens of metres on a bad day.
- It is a far-field law. Close to any source bigger than a fist the sphere has not formed yet. For a single cabinet that means within a metre or two; for a line array it means the whole near field, which is the next section.
- Boundaries add, they do not subtract. A source on the ground radiates into a half-sphere and is 6 dB up on its free-field figure; against a wall and a floor, 12. A subwoofer on the deck is not breaking the law, it is radiating into half the space.
Line sources: three decibels a doubling, up to a point
A tall, continuous line of sources radiates a cylindrical wave in its near field. The wavefront's area grows with the radius rather than its square, so the level falls at 3 dB per doubling — and that is the entire case for the line array. The catch is where the near field ends. Urban, Heil and Bauman's Fresnel analysis gives the border distance as
dB = 1.5 · F · H² · √(1 − 1 / (3 F H)²)
with H the array height in metres and F the frequency in kilohertz. Three things follow, and each is checked against the paper's own numbers at build time:
- The near field is frequency-dependent. Roughly, it extends in proportion to frequency: for the paper's 5.4 m example it reaches 88 m at 2 kHz, and a quarter of that at 500 Hz. The same array is a line source for the top end and a point source for the bottom, from the same listening position.
- It grows with the square of the height. Doubling the array quadruples the near field. That is why length matters more than box count, and why a short array is a loud point source with extra steps.
- Below 1 / (3H) kilohertz there is no near field at all. A 4 m array radiates spherically from the first metre at anything under about 80 Hz, which is why the subs hung with it gain nothing from the geometry and are a separate design problem.
| 6 m array, loss from 1 m — frequency ↓ distance → | 4 m | 8 m | 16 m | 32 m | 64 m | 128 m | Near field ends |
|---|---|---|---|---|---|---|---|
| 125 Hz | −6.0 dB6.0 dB better than a point | −10.2 dB7.8 dB better than a point | −16.3 dB7.8 dB better than a point | −22.3 dB7.8 dB better than a point | −28.3 dB7.8 dB better than a point | −34.3 dB7.8 dB better than a point | 6 m |
| 250 Hz | −6.0 dB6.0 dB better than a point | −9.0 dB9.0 dB better than a point | −12.9 dB11.2 dB better than a point | −18.9 dB11.2 dB better than a point | −24.9 dB11.2 dB better than a point | −31.0 dB11.2 dB better than a point | 13 m |
| 500 Hz | −6.0 dB6.0 dB better than a point | −9.0 dB9.0 dB better than a point | −12.0 dB12.0 dB better than a point | −15.8 dB14.3 dB better than a point | −21.8 dB14.3 dB better than a point | −27.9 dB14.3 dB better than a point | 27 m |
| 1 kHz | −6.0 dB6.0 dB better than a point | −9.0 dB9.0 dB better than a point | −12.0 dB12.0 dB better than a point | −15.1 dB15.1 dB better than a point | −18.8 dB17.3 dB better than a point | −24.8 dB17.3 dB better than a point | 54 m |
| 2 kHz | −6.0 dB6.0 dB better than a point | −9.0 dB9.0 dB better than a point | −12.0 dB12.0 dB better than a point | −15.1 dB15.1 dB better than a point | −18.1 dB18.1 dB better than a point | −21.8 dB20.3 dB better than a point | 108 m |
| 4 kHz | −6.0 dB6.0 dB better than a point | −9.0 dB9.0 dB better than a point | −12.0 dB12.0 dB better than a point | −15.1 dB15.1 dB better than a point | −18.1 dB18.1 dB better than a point | −21.1 dB21.1 dB better than a point | 216 m |
| 8 kHz | −6.0 dB6.0 dB better than a point | −9.0 dB9.0 dB better than a point | −12.0 dB12.0 dB better than a point | −15.1 dB15.1 dB better than a point | −18.1 dB18.1 dB better than a point | −21.1 dB21.1 dB better than a point | 432 m |
Loss relative to one metre on axis of a 6 m flat line source, cylindrical out to the border distance and spherical beyond it. The one-metre reference is a convenience for comparison: a real array's published sensitivity is extrapolated from the far field, not measured at a metre. The array's height is also its only geometry here — a curved array trades near-field reach for vertical coverage, and the paper covers that too.
Air absorption
Air is not a perfect spring. Part of the energy in a sound wave goes into making oxygen and nitrogen molecules vibrate, and a molecule that is still vibrating when the pressure reverses gives its energy back late — as heat, not sound. How much is lost depends on how fast the molecules relax compared with the frequency, and the relaxation rate depends on how much water vapour is present, because water collisions are what let the molecules give the energy up. That is the whole physics of ISO 9613-1, and it is implemented here in full rather than interpolated from a chart.
The result is an attenuation coefficient, in decibels per metre, that is added on top of the geometric loss. It rises roughly with the square of frequency: negligible below 500 Hz at any sensible distance, a few decibels per hundred metres at 4 kHz, and a genuine wall at 16 kHz.
The humidity dependence is not the direction intuition suggests. At 20 °C and 8 kHz the absorption peaks at 17 % relative humidity (225 dB/km) and is only 63 dB/km at 90 %. Dry air is the enemy of the top end, which is why a desert festival at dusk loses its air and a humid summer night keeps it.
| dB per km — conditions ↓ band → | 63 | 125 | 250 | 500 | 1 k | 2 k | 4 k | 8 k | 16 k * |
|---|---|---|---|---|---|---|---|---|---|
| 10 °C, 70 %Cool, damp | 0.12 | 0.41 | 1.04 | 1.92 | 3.66 | 9.70 | 33 | 118 | 370 |
| 20 °C, 70 %Mild, damp | 0.09 | 0.33 | 1.12 | 2.79 | 4.98 | 9.04 | 23 | 78 | 281 |
| 20 °C, 50 %Mild, average | 0.12 | 0.44 | 1.31 | 2.73 | 4.66 | 9.89 | 30 | 105 | 365 |
| 20 °C, 20 %Mild, dry | 0.26 | 0.71 | 1.39 | 2.59 | 6.53 | 22 | 75 | 217 | 435 |
| 30 °C, 70 %Hot, humid | 0.07 | 0.25 | 0.95 | 3.12 | 7.41 | 13 | 23 | 60 | 203 |
| 30 °C, 20 %Hot, dry | 0.21 | 0.72 | 1.86 | 3.40 | 6.00 | 15 | 48 | 167 | 514 |
Pure-tone absorption coefficients in dB per kilometre at sea-level pressure, per ISO 9613-1. Divide by ten for dB per 100 m. * 16 kHz is beyond the standard's stated range.
Both at once: what reaches the back
Put the two together for a point source at 20 °C and 50 % RH. The spreading loss is one number per distance. The air loss is one number per distance per frequency, and it is the second column that changes the tonal balance.
| from 1 m — distance ↓ | Spreading | Air at 1 k | Air at 4 k | Air at 8 k | Air at 16 k | Total at 8 k |
|---|---|---|---|---|---|---|
| 10 m | −20.0 | −0.0 | −0.3 | −0.9 | −3.3 | −20.9 |
| 20 m | −26.0 | −0.1 | −0.6 | −2.0 | −6.9 | −28.0 |
| 50 m | −34.0 | −0.2 | −1.5 | −5.2 | −17.9 | −39.1 |
| 100 m | −40.0 | −0.5 | −2.9 | −10.4 | −36.1 | −50.4 |
| 200 m | −46.0 | −0.9 | −5.9 | −21.0 | −72.5 | −67.0 |
- At a hundred metres the air costs more at 8 kHz than a whole doubling of distance does. The spreading loss is the same at 1 kHz and 8 kHz; the extra 10 dB at 8 kHz is pure tilt. This is what "it sounds dull at the back" is, in numbers, and no amount of gain fixes a tilt.
- Delay towers exist because of the second column, not the first. Another 6 dB of level could be found at the stage. Another 10 dB of 8 kHz could not: it would have to pass through the same 100 m of air. A source placed 60 m closer restores the top end because it shortens the path, which is the one thing EQ cannot do.
- Prediction software's "air absorption compensation" is a high-frequency shelf derived from exactly this coefficient and the throw distance — and it is a compromise, because the boost that flattens the back row is the same boost that hardens the front. The array's frequency-dependent near field, above, is the other half of why the top end reaches further than the simple table suggests.
Your distance, your weather
The same two models with your numbers in them. Give the level at a reference distance, the distance you care about and the conditions; choose a point source or a flat line source of a stated height. The per-band rows separate what the geometry took from what the air took, because they are fixed by different things.
| Band | Spreading | Air | Level | Near field |
|---|---|---|---|---|
| 250 Hz | — | — | — | — |
| 500 Hz | — | — | — | — |
| 1 kHz | — | — | — | — |
| 2 kHz | — | — | — | — |
| 4 kHz | — | — | — | — |
| 8 kHz | — | — | — | — |
| 16 kHz * | — | — | — | — |
This calculator needs JavaScript. The tables above cover the same models at fixed distances and conditions.
Free field, on axis, sea-level pressure, no wind, no ground and no room. Those are not small omissions: over a hundred metres outdoors a temperature inversion or a following wind can move the answer by more than the air absorption does, in either direction. Treat the output as the floor that physics guarantees, not as a prediction. * 16 kHz is outside ISO 9613-1's stated range.
Sources
- ISO 9613-1:1993, Acoustics — Attenuation of sound during propagation outdoors — Part 1: Calculation of the absorption of sound by the atmosphere. The relaxation-frequency model implemented above, equations (3) to (5) and Annex B. Paywalled; the formulation is reproduced in many places, including SONAR.m's documentation, with the same constants used here.
- ISO 9613-2:1996 Table 2 — the octave-band coefficient row for 10 °C and 70 % RH that the build asserts against, as reproduced by EMD in the windPRO DECIBEL appendix. EMD note the table was in earlier editions of Part 2 and not the current one; the values are Part 1's formula evaluated, which is what makes them a fair test of it.
- Urban, Heil and Bauman, "Wavefront Sculpture Technology", J. Audio Eng. Soc. vol. 51 no. 10, October 2003, pp. 912–932. The border-distance formula and the 5.4 m / 88 m / 2 kHz example are from §1; L-Acoustics publish the paper. The paper's Fresnel treatment in §4 places the border at half the distance the formula gives; this page uses the formula, which the authors describe as a reasonable average of the geometric and numerical expressions.
- Geometric spreading — conservation of energy over a sphere or a cylinder. There is no source to cite for it that is better than deriving it.
Assembled 8 September 2026 with AI assistance. Nothing here has been measured on a field; the two models are validated against the published figures named above and are only as good as those figures. The line-source model is a flat, continuous source on axis, which no real array is exactly, and the air model is pure-tone absorption in still, uniform air, which no real afternoon is.
Companion pages: equal-loudness contours for why the level that reaches the back row also changes what it sounds like, and polarity, phase and time alignment for what to do with the delay tower once it exists.