Why decibels cannot be added directly
A decibel is not a quantity of sound; it is a logarithm of a ratio. A sound pressure level of 85 dB means the mean-square pressure is 108.5 times the reference of 20 micropascals squared. Logarithms turn multiplication into addition, which is exactly why they are convenient - and exactly why adding the logarithms of two sources gives you the logarithm of their product, which is meaningless here.
What actually adds is energy. Two independent sources deliver twice the acoustic energy of one, and the level corresponding to twice the energy is 10 x log10(2) = 3.0103 dB more. That single number governs the whole subject: every doubling of the number of identical sources adds 3.01 dB, and no amount of doubling ever adds more. Ten sources add 10 dB, a hundred add 20 dB, a thousand add 30 dB.
The consequence that surprises people is how little a quieter source matters. A source 10 dB below the loudest contributes 10 x log10(1 + 0.1) = 0.41 dB. A source 20 dB below contributes 0.04 dB, which is below the resolution of most meters. If you are trying to reduce a noise problem, this is the most useful fact on the page: silencing anything more than about 10 dB below the dominant source is effort spent for nothing.
The formula, and where the 10 and the 20 come from
Convert each level back to a relative energy with 10L/10, add those numbers, and convert the sum back with 10 x log10. That is the whole calculation. Nothing about it depends on the sources being of the same type, and it applies equally to sound power levels, sound pressure levels and electrical power levels - anything defined with a 10 log convention.
The choice between 10 log and 20 log is a persistent source of confusion, and it is not arbitrary. Power-like quantities - power, intensity, mean-square pressure, energy - use 10 log. Amplitude-like quantities - voltage, sound pressure itself, current - use 20 log, because power goes as amplitude squared and log(x2) = 2 log(x). A sound pressure level is already defined through the squared pressure, so it is power-like and combines with 10 log. This is why doubling the number of sources gives 3.01 dB while doubling the voltage into a resistor gives 6.02 dB.
Distance follows from the same reasoning. A point source radiating into free space spreads its power over a sphere whose area grows as the square of the radius, so intensity falls as 1/r2 and the level falls by 20 x log10(r2/r1) - which is 6.02 dB per doubling of distance. An idealised infinite line source spreads over a cylinder whose area grows only linearly with radius, so the fall is 10 x log10(r2/r1), or 3.01 dB per doubling. ISO 9613-2 treats geometric spreading this way and then adds terms for air absorption, ground effect and screening, all of which this calculator excludes.
One important assumption underlies all of it: the sources must be uncorrelated. Independent machines, separate instruments and a band on stage all qualify. Two loudspeakers fed the same signal do not - they are coherent, and coherent sources add in pressure rather than in energy, giving up to 6.02 dB where they arrive in phase and cancelling where they arrive in antiphase. The delay and reverb time calculator covers the timing side of that interference.
Worked example: four machines in a workshop
Four machines are measured individually at the operator's position: 85, 82, 78 and 75 dB. What is the total, and what would silencing each one achieve?
- Convert each to relative energy. 108.5 = 316,227,766; 108.2 = 158,489,319; 107.8 = 63,095,734; 107.5 = 31,622,777.
- Add them. 316,227,766 + 158,489,319 + 63,095,734 + 31,622,777 = 569,435,596.
- Convert back. 10 x log10(569,435,596) = 10 x 8.75536 = 87.55 dB.
- Compare with the loudest alone. 87.55 − 85 = 2.55 dB gained from the other three combined.
- Shares. The 85 dB machine is 316,227,766 / 569,435,596 = 55.5% of the energy; the 82 dB machine is 27.8%; the 78 dB machine 11.1%; the 75 dB machine 5.6%.
- Silence the quietest. The remaining sum is 537,812,819, giving 10 x log10(537,812,819) = 87.31 dB - a saving of just 0.25 dB, which nobody can hear.
- Silence the loudest instead. The remaining sum is 253,207,830, giving 84.03 dB - a saving of 3.52 dB, fourteen times as much benefit for the same piece of work.
- Take the total out to 10 m. If those levels were measured at 1 m and the machines behave as point sources, the correction is −20 x log10(10/1) = −20 dB, giving 67.55 dB at 10 m.
Step 7 is the point of the whole exercise. Noise control money goes to the dominant source, and the share column tells you which that is far more clearly than the raw levels do.
Reading the increase figure
The increase over the loudest source is the number that tells you whether the other sources matter. If it is under 0.5 dB, the problem is one source and everything else is noise in the accounting. If it is close to 3 dB you have two comparable sources and must deal with both. If it is close to 6 dB you have four comparable sources, and to 10 dB you have ten.
Translating decibels into perception needs care and honest hedging. A 3 dB change is at the edge of noticeable on a steady sound heard alone, though it is easy to hear as an A/B comparison. A 10 dB change is conventionally described as a doubling or halving of loudness, a rule of thumb from loudness scaling experiments rather than a physical fact - and it corresponds to ten times the acoustic energy. That gap between energy and perception is why turning off nine of ten identical machines only halves the perceived loudness.
Two cautions about comparing a computed total against a limit. First, weighting: occupational and environmental criteria are defined on A-weighted levels, which de-emphasise low frequencies to approximate the ear's response at moderate levels. An unweighted total and an A-weighted limit are different quantities and cannot be compared. Second, time: exposure standards are stated as time-weighted averages over a shift, not as instantaneous levels, so a momentary figure from this calculator is not directly a compliance number.
For distance, remember the correction only covers geometric spreading. Outdoors, air absorption removes additional energy that grows with distance and strongly with frequency; ground and barriers change things further. Indoors, the reverberant field dominates beyond the critical distance, and beyond that point moving further away barely reduces the level at all - which is why the inverse square law usually fails badly in a real room.
How much a second source adds
| Difference between the two levels (dB) | Add to the higher level (dB) | Share of the quieter source (%) |
|---|---|---|
| 0 | 3.01 | 50.0 |
| 1 | 2.54 | 44.3 |
| 2 | 2.12 | 38.7 |
| 3 | 1.76 | 33.4 |
| 4 | 1.46 | 28.5 |
| 5 | 1.19 | 24.0 |
| 6 | 0.97 | 20.1 |
| 8 | 0.64 | 13.7 |
| 10 | 0.41 | 9.1 |
| 15 | 0.14 | 3.1 |
| 20 | 0.04 | 1.0 |
Read the row for 3 dB as: two sources at 90 and 87 dB combine to 90 + 1.76 = 91.76 dB, and the quieter one accounts for a third of the total energy. Beyond a 10 dB gap the addition is within the repeatability of most field measurements.
Mistakes that give the wrong total
- Adding the decibel figures arithmetically. 85 + 85 is not 170. It is 88.01, because the energies add and the logarithm does not.
- Averaging levels arithmetically. The mean of 90 and 70 dB is not 80 dB in energy terms - the energy average is 10 x log10((10^9 + 10^7)/2) = 87.03 dB, because the louder value dominates completely.
- Using 10 log where 20 log belongs, or the reverse. Power-like quantities use 10 log; amplitude-like quantities use 20 log. Sound pressure level is already power-like and takes 10 log.
- Combining coherent sources with this formula. Two speakers on the same signal interfere rather than add energetically, giving up to 6 dB in phase and deep cancellation out of phase.
- Mixing weightings. A-weighted and unweighted levels are different quantities. Combine like with like, and only compare the total against a limit expressed in the same weighting.
- Applying the distance correction indoors. Beyond the critical distance a room's reverberant field dominates and the level stops falling with distance in anything like the free-field way.
- Combining levels measured at different positions. Every level in the sum must be what that source produces at the one position you care about.
Where decibel addition turns up
In live sound, it is the arithmetic behind array design. Doubling the number of cabinets in a stack adds 3 dB of level if the boxes are driven independently and their outputs are uncorrelated, but a properly coupled array in the frequency range where the boxes are close together relative to a wavelength behaves coherently and can gain up to 6 dB. Which regime you are in depends on frequency and spacing, which is why line arrays behave so differently at 100 Hz and 5 kHz. The speaker impedance calculator covers the electrical side of adding cabinets, and it is worth noting that doubling amplifier power also gains 3 dB - the same logarithm from a different cause.
In occupational hygiene, energy summation underlies the equivalent continuous level, Leq, which averages a fluctuating exposure by energy rather than by level. That is why a few loud moments dominate a shift average, and why a 3 dB exchange rate - halving the permitted time for every 3 dB - is the energy-consistent choice. Some regulators use a 5 dB exchange rate for historical reasons, which is not energy-consistent, so check which applies before converting anything.
In environmental acoustics, the same summation combines traffic, plant and rail contributions at a receptor, each first corrected for its own distance and screening. That is where the point-versus-line distinction earns its keep: a single machine spreads spherically at 6 dB per doubling, while a busy road behaves as a line at 3 dB per doubling, so a road stays audible much further away than its peak level would suggest.
And in studio work, it explains why a mix's level does not rise the way the track count suggests. Adding a second uncorrelated track raises the sum by 3 dB, a fourth by another 3, and so on - so a 32-track mix of similar-level uncorrelated material sits about 15 dB above one track, not 32 times higher. Correlated material, such as a doubled vocal, behaves quite differently and can add up to 6 dB.