Sound level composition

The decibel is a logarithmic scale: add two equal sound levels and you do not get double the number, only 3 dB more. It is the most common mistake of anyone new to acoustics — two machines at 60 dB(A) each do not give 120 dB(A), they give 63 dB(A). This Open Lab tool performs the correct composition of several levels, band summation with weighting, and the subtraction of background noise from a measurement.

Why levels do not add arithmetically

The level in decibels is ten times the logarithm of the ratio between an energy-like quantity (sound pressure squared) and a reference. What adds up is the energies, not the levels. Doubling the energy means adding 10·log₁₀(2) ≈ 3.01 dB; multiplying it tenfold means adding 10 dB. Two practical rules follow: two identical sources give +3 dB compared with one; a source 10 dB weaker than another contributes less than half a decibel to the total.

The 10·log and 20·log trap

The same level is written in two ways, and confusing them is the mistake that immediately follows the previous one:

$$L_p=10{\log }_{10}\left(\frac{p^2}{p_0^2}\right)=20{\log }_{10}\left(\frac{p}{p_0}\right)$$

The two expressions are the same thing — the square leaves the logarithm as a factor of 2 — but they signal which quantity you are handling. 10·log applies to squared quantities (energy, power, intensity, p²), 20·log to amplitudes (pressure, voltage). Summing sources happens on energies, so it always goes through 10·log: adding two levels as if they were amplitudes (10^(L/20)) overestimates the result. With two 60 dB sources the correct figure is 63.0 dB; treated as amplitudes they would give 66.0 dB.

The reference p₀ = 20 µPa never enters the sum: it cancels. This is why composing levels does not require knowing which reference they are expressed against, provided it is the same for all of them.

The correction table every engineer keeps in mind

A practical rule for combining two levels follows from these relations: given the gap ΔL between them, how much to add to the larger one to obtain the sum.

The curve shows at a glance why the loudest source dominates in acoustics: as soon as two levels are 10 dB apart, the weaker one is worth four tenths of a decibel in the total — below the repeatability of an ordinary measurement. Reducing a source already dominated by another achieves nothing until the dominant one is brought down first.

The energetic sum

For N measured levels Lᵢ the overall level is the logarithmic sum of the energies:

$$L_{\mathrm{tot}}=10{\log }_{10}\left(\sum _{i}10^{L_i/10}\right)$$

The same formula covers two distinct situations: composing the contributions of several sources operating simultaneously (fan + compressor + traffic) to obtain the level at the receiver, or reconstructing the broadband overall level from separately measured frequency bands — for instance summing the octaves from 63 Hz to 8 kHz. Alongside the total, the tool shows each source’s percentage contribution to the overall energy: it is the most direct way to see where a noise reduction is worth spending on.

A worked example

Three sources contribute to the level at a point: fan 72 dB(A), compressor 68 dB(A), traffic 65 dB(A). The relative energies are 10^7.2, 10^6.8 and 10^6.5; their sum corresponds to a total level of 74.0 dB(A). Unpacking the percentage contributions: the fan accounts for 62.6 % of the energy, the compressor 24.9 %, the traffic only 12.5 %.

From here you can read the return on each intervention, and the reading is always sobering:

Silencing the traffic completely — the most conspicuous intervention on paper — buys barely half a decibel. But even a 3 dB cut on the dominant source returns less than one hopes: −1.6 dB on the total, not −3, because the rest of the energy stays where it was. This is the most frequent disappointment in noise-control work: a silencer is sized for the worst machine, the machine delivers the designed cut, and at the receiver you measure half of it. The numbers only add up when several sources are treated together, as the last row shows.

Band summation and weighting

The most frequent case is not composing different sources but recomposing a spectrum. An octave-band sound level meter returns ten numbers; the overall level in dB(A) is obtained by applying the A-weighting to each band and then summing in energy — in that order. Weighting after the sum gives a different and meaningless number.

The A-weighting values at the nominal octave centres, from the transfer function in IEC 61672-1, are:

The shape explains why A-weighting is so often contested at low frequencies: at 31.5 Hz it removes almost 40 dB. A fan pushing all its energy below 63 Hz can look acceptable in dB(A) and be unbearable in the next room — which is why low-frequency noise criteria use C-weighting or band analysis rather than dB(A) alone.

$$L_A=10{\log }_{10}\left(\sum _{b}10^{(L_{Z,b}+A_b)/10}\right)$$

A recomposition example

An unweighted (Z) spectrum, typical of a rotating machine with its energy concentrated low down:

The unweighted overall level is 91.1 dB, the A-weighted one 76.2 dB(A): nearly fifteen decibels of difference, all of it from the two lowest octaves, which alone carry 80 % of the unweighted energy and become irrelevant in dB(A). The bands that govern the dB(A) figure are 250, 500 and 1000 Hz — and that is where the treatment must be sized if the constraint is expressed in dB(A).

Background-noise correction

When measuring a source, the sound level meter always records the ambient background as well. To isolate the source’s contribution you subtract the background energy from the measured energy:

$$L_{\mathrm{corr}}=10{\log }_{10}\left(10^{L_{\mathrm{mis}}/10}-10^{L_{\mathrm{bg}}/10}\right)$$

The normative form is the same thing written as a correction to subtract from the reading, a function of the gap ΔL = L_mis − L_bg alone:

$$K_1=-10{\log }_{10}\left(1-10^{-\Delta L/10}\right),L_{\mathrm{corr}}=L_{\mathrm{mis}}-K_1$$

Which standard sets the thresholds. The formula is the same throughout acoustics; the scope is not. The thresholds used here — correction negligible above a 10 dB gap, applicable between 3 and 10, result unreliable below — are those of ISO 1996-2, the environmental noise standard. The sound power standards, ISO 3744 and ISO 3741, apply the same subtraction with their own, stricter requirements on the background margin, because there a machine is being qualified under controlled conditions. A report cites the standard of the field being measured: citing 3744 for an environmental measurement is a reference that does not hold once the other party opens it.

Written this way the stakes are immediately visible: K₁ is 0.46 dB at ΔL = 10 dB, 1.26 dB at 6 dB, 2.20 dB at 4 dB and 3.02 dB at 3 dB. The correction diverges as the background approaches the measurement, and it diverges fast.

Why the threshold sits at 3 dB

The reason is not conventional: it is uncertainty propagation. Differentiating L_corr with respect to the two readings and setting r = 10^(−ΔL/10) gives, for equal uncertainties u on measurement and background:

$$u(L_{\mathrm{corr}})=u\cdot \frac{\sqrt{1+r^2}}{1-r}$$

With a typical uncertainty of ±0.5 dB on each reading, at ΔL = 10 dB the corrected result carries ±0.56 dB — practically nothing new. At ΔL = 3 dB it carries ±1.12 dB, an uncertainty worth half the correction itself. At ΔL = 1 dB it carries ±3.1 dB: the computed number is formally defined and operationally empty. The 3 dB threshold is the point where the uncertainty doubles; the method for building that budget is in the measurement uncertainty (GUM) wiki.

Current practice therefore distinguishes three cases:

$$\{\begin{array}{ll}\Delta L>10\mathrm{dB}& \text{correction negligible,}{L}_{\mathrm{corr}}\approx L_{\mathrm{mis}}\\ 3\le \Delta L\le 10& \text{correction applicable}\\ \Delta L<3\mathrm{dB}& \text{unreliable: upper bound}{L}_{\mathrm{corr}}\le L_{\mathrm{mis}}-3\mathrm{dB}\end{array}$$

One caveat: these are general-practice margins. Product standards are stricter, and the required margin depends on the method. ISO 3741, for determining sound power in a reverberation room, requires the background to sit at least 6 dB below in the bands at the edges of the range and at least 10 dB below between 250 Hz and 5 kHz; ISO 16283-1, for field measurements of building sound insulation, requires at least 6 dB per band, preferably more than 10. Before applying the 3 dB rule, check that the method being followed admits it.

Below 3 dB of difference the source is submerged in the background: the meter no longer separates the two contributions. In that case the tool does not invent a value: it returns an upper bound and flags it explicitly.

A subtraction example

With the source running the meter reads 75 dB(A); with the source off, the background alone is 69 dB(A). The difference is ΔL = 6 dB, so the correction is applicable: subtracting the energies gives 73.7 dB(A), a correction of −1.3 dB from the raw reading. If instead the background were 73 dB(A) (ΔL = 2 dB), the formula would give ~70.7 dB(A) but the result is unreliable: only the upper bound L_mis − 3 = 72 dB(A) is reported, because at that distance from the background the measurement no longer isolates the source alone.

A, C, Z weightings

Every level carries a spectral weighting with it (IEC 61672-1). A approximates the ear’s response at moderate levels and is the standard weighting for disturbance assessment and for most Italian and European regulation. C has a flatter response and is used for the peak level and for low-frequency phenomena. Z is the absence of weighting, a linear response between 10 Hz and 20 kHz.

Levels with different weightings are not directly addable: composing an L_A with an L_Z produces a mathematically correct but physically meaningless number. The tool still performs the sum on request, but shows a warning.

The difference L_C − L_A, on the other hand, is a useful indicator precisely because the two weightings diverge low down: on the example spectrum above it is about 13 dB, and a gap of that order signals that the energy sits in the low frequencies and that dB(A) alone will describe the perceived disturbance poorly.

Limits of the method

• The logarithmic sum assumes incoherent sources. Two loudspeakers reproducing the same signal are coherent sources: in phase they give +6 dB instead of +3 (amplitudes add, not energies), in antiphase they cancel. The energy model describes neither case.
• Background subtraction assumes the background noise is stationary and representative of the measurement interval. A background that changes between the with-source and without-source measurements makes the subtraction meaningless regardless of ΔL.
• For ΔL < 3 dB the value returned is a bound, not a measurement.
• The A-weighting values tabulated here are at the nominal octave centres. For a spectrum that is very steep within a band, the centre value does not represent the whole band and the recomposition should be done in third-octaves.

A demonstrative tool: it does not replace the measurements the applicable methodology requires, nor the professional responsibility of whoever signs a conformity assessment. “Certified software” is not a category Italian regulation provides for these calculations: what counts is that the model be adequate to the method and verifiable.

Normative references

• ISO 1996-2:2017 — Description, measurement and assessment of environmental noise: determination of levels. The applicability thresholds used on this page and in the tool come from it.
• ISO 3744:2025 — Determination of sound power levels of noise sources, engineering methods for an essentially free field over a reflecting plane. It is the edition in force and supersedes ISO 3744:2010, withdrawn. A caveat on this page’s citations: the clause numbering (Annex A.3) and the ΔL thresholds quoted come from the 2010 edition, and we do not transfer them to 2025 by analogy — they need checking against the licensed text of the new edition.
• ISO 3741:2010 — Determination of sound power levels in reverberation rooms: minimum required background-noise margins per band.
• ISO 16283-1:2014 — Field measurement of sound insulation in buildings: minimum 6 dB per band margin on background noise.
• UNI 9884:1997 — Acoustic characterisation of the territory through the description of environmental noise. Withdrawn without replacement: cited here only for historical continuity, not as a current reference.
• IEC 61672-1:2013 — Sound level meters: definition and transfer function of the A, C and Z weightings.
• JCGM 100:2008 (GUM) — Expression of uncertainty in measurement: propagation applied to the background correction.

The tool is available as a runnable instrument at /en/tools/level-composition/.