Why sound is measured on a logarithmic scale
Human hearing spans an enormous dynamic range. The quietest audible sound at 1 kHz corresponds to a pressure of about 20 micropascals; a jet engine at close range reaches around 200 pascals. That is a factor of ten million in pressure and a hundred trillion in intensity. Writing those numbers linearly is unusable, so acoustics compresses them logarithmically into decibels.
A decibel is not a unit of sound. It is one tenth of a bel, and a bel is the base-10 logarithm of a power ratio. Every decibel figure therefore needs a reference, and for airborne sound the international references fixed by ISO 1683 are 10−12 W/m² for intensity and 20 µPa for pressure. The pressure reference was chosen because it is close to the threshold of hearing at 1 kHz, which is what makes 0 dB SPL mean roughly the quietest audible sound rather than silence.
Intensity is proportional to the square of pressure, so a level defined on pressure needs a factor of 20 where one defined on intensity needs 10: 20 log₁₀(p/p₀) against 10 log₁₀(I/I₀). The two references are chosen so the numbers agree. With the conventional reference impedance ρc = 400 Pa·s/m, p₀²/ρc = (2×10⁻⁵)²/400 = 10⁻¹² exactly, which is I0. Real air at 20 °C and standard pressure has ρc ≈ 413, so a rigorous treatment puts SPL about 0.14 dB above intensity level. That difference is smaller than any field measurement uncertainty, and this calculator uses 400 for consistency.
Two rules of thumb follow from the logarithm and are worth memorising. +3 dB doubles the power, because 10 log 2 = 3.01. +6 dB doubles the pressure, because 20 log 2 = 6.02. And roughly +10 dB is heard as twice as loud, which is a psychoacoustic result, not an arithmetic one.
Distance, and why ten machines are not ten times louder
A point source radiating freely spreads its power over a sphere whose area grows as 4πr². Intensity therefore falls as the inverse square of distance, and the level falls by 20 log₁₀(r₂/r₁). Every doubling of distance removes 6.02 dB. Ten metres from a source is 20 dB quieter than one metre; a hundred metres is 40 dB quieter.
Three qualifications matter outdoors and indoors. A source sitting on a hard reflective surface radiates into a hemisphere rather than a sphere, halving the area and adding about 3 dB. A long line source such as a motorway behaves differently again: its level falls by only 3 dB per doubling, because the area grows as 2πrL. And inside a room, the level stops falling once you leave the direct field and enter the reverberant field, where reflected energy dominates and the level is set by the room's absorption rather than by distance.
Adding sources works on power, never on level. Two independent machines each producing 85 dB give 10 log₁₀(10^8.5 + 10^8.5) = 88.01 dB, not 170. Ten of them give 95 dB. A hundred give 105. The general rule for n equal incoherent sources is +10 log₁₀(n), so the marginal contribution of each new machine shrinks steadily.
The word incoherent is load-bearing. Independent sources add in power because their pressures have random relative phase, and the cross term averages to zero. Two loudspeakers fed the identical signal are coherent, and where their pressures arrive in phase they add linearly, giving +6 dB rather than +3; where they arrive out of phase they cancel. That is why a stereo pair produces comb filtering and why an anti-noise system can subtract sound entirely.
Worked example: a compressor on a factory roof
A compressor radiates 0.5 acoustic watts. It sits on a large flat roof, so it radiates into a hemisphere. What level does a worker measure at 3 m, and what happens if three more identical units are installed?
- Hemispherical area at 3 m.
A = 2πr²= 2 × 3.14159 × 9 = 56.549 m². - Intensity. I = 0.5 ÷ 56.549 = 8.8419 × 10−3 W/m².
- Level. L = 10 log10(8.8419×10−3 ÷ 10−12) = 10 log10(8.8419×109) = 10 × 9.94654 = 99.47 dB.
- Pressure. p = √(I × 400) = √3.5368 = 1.8806 Pa. Check: 20 log10(1.8806 ÷ 2×10−5) = 20 log10(94,030) = 99.47 dB. The two routes agree.
- Add three more units. Four equal incoherent sources add
10 log₁₀(4)= 6.02 dB, giving 105.49 dB. Quadrupling the machinery costs 6 dB, not 400%. - Move away. At 12 m instead of 3 m — two doublings — the four-unit level drops by 20 log10(4) = 12.04 dB to 93.45 dB. That is an unweighted figure; for a broadband compressor the A-weighted level is typically a few decibels lower, and it is the A-weighted number that has to be compared with the 85 dBA limit.
If you reproduce this in the calculator, note that its power mode assumes a full sphere. Enter 0.5 W at 3 m and it returns 96.46 dB; add 10 log10(2) = 3.01 dB for the hemispherical mounting to recover the 99.47 dB above.
To get below 85 dB purely by distance you would need 20 log₁₀(r/3) = 20.49, so r = 3 × 101.0245 = 31.8 m. That calculation is why noise control engineers attack the source or add a barrier rather than relying on separation.
Reading the number: weighting, exposure limits and what is loud
The level this calculator returns is unweighted — a flat, physical decibel figure. Occupational and environmental limits are almost always quoted in dBA, which applies a frequency weighting that de-emphasises bass to mimic the ear's reduced sensitivity at low frequencies. A-weighting subtracts roughly 26 dB at 63 Hz, 9 dB at 250 Hz, 3 dB at 500 Hz, nothing at 1 kHz, and adds about 1 dB near 2–4 kHz. If your source is broadband, the unweighted level is generally the higher number.
For exposure, two US limits sit side by side. The NIOSH recommended exposure limit is 85 dBA for an 8-hour time-weighted average with a 3 dB exchange rate, meaning the permitted time halves for every 3 dB — 4 hours at 88 dBA, 2 hours at 91, 15 minutes at 100. The OSHA permissible exposure limit is 90 dBA for 8 hours with a 5 dB exchange rate, which is more permissive at high levels. Around 120 dB most listeners report pain, and impulsive peaks above 140 dB can damage hearing instantly regardless of duration.
Sanity-check the magnitude against everyday anchors: a quiet bedroom at night is 25–30 dB, normal conversation at 1 m is around 60 dB, a busy street is 75–85 dB, and a chainsaw at the operator's ear is around 105–110 dB. If your computed level falls outside 0–150 dB you have almost certainly entered intensity where pressure was expected, or watts where milliwatts were meant.
Decibel level, intensity and pressure side by side
| Level (dB) | Intensity (W/m²) | Pressure (Pa) | Familiar comparison |
|---|---|---|---|
| 0 | 1 × 10⁻¹² | 0.00002 | Threshold of hearing at 1 kHz |
| 20 | 1 × 10⁻¹⁰ | 0.0002 | Rustling leaves, a broadcast studio |
| 40 | 1 × 10⁻⁸ | 0.002 | Quiet library, refrigerator hum |
| 60 | 1 × 10⁻⁶ | 0.02 | Conversation at one metre |
| 80 | 1 × 10⁻⁴ | 0.2 | Kerbside traffic, alarm clock |
| 85 | 3.162 × 10⁻⁴ | 0.3557 | NIOSH 8-hour recommended limit |
| 94 | 2.512 × 10⁻³ | 1.0024 | Acoustic calibrator reference (1 Pa) |
| 100 | 1 × 10⁻² | 2.0 | Chainsaw, nightclub dance floor |
| 120 | 1 | 20 | Threshold of pain, siren at close range |
| 140 | 100 | 200 | Jet engine at 25 m; instant damage risk |
Intensity is I₀ × 10^(L/10) and pressure is p₀ × 10^(L/20), evaluated at each level. The comparisons are typical A-weighted field figures and vary with source and distance.
Errors that produce a wrong decibel figure
- Adding decibels arithmetically. 60 dB plus 60 dB is 63 dB, not 120 dB. Convert each level to power, sum, then convert back.
- Using 10 log where 20 log belongs. Pressure, voltage and any other amplitude quantity take the factor 20; power, intensity and energy take 10. Mixing them halves or doubles the answer.
- Confusing sound power with sound pressure level. Sound power level (LW, referenced to 1 pW) is a property of the machine and does not depend on distance; sound pressure level depends on where you stand. A fan rated at 80 dB LW does not produce 80 dB at your ear.
- Applying the inverse-square law indoors. Beyond the critical distance the reverberant field dominates and the level barely falls. Predicting a 40 dB drop across a factory floor will disappoint you.
- Reading an amplifier's electrical watts as acoustic watts. Loudspeaker efficiency is typically 0.5–2%, so 100 electrical watts is one or two acoustic watts at best.
- Comparing an unweighted level with a dBA limit. Exposure standards are written in dBA. A bass-heavy source can be 10 dB higher unweighted than A-weighted.
- Assuming two speakers always add 3 dB. That holds for incoherent sources. Fed the same signal, they can add 6 dB where they are in phase and cancel where they are not.
Related quantities and when to use a different tool
Sound level answers how much energy is arriving, not what is arriving. For frequency content you need a spectrum, and for tonal problems the relevant quantity is the frequency itself, which connects to the wavelength and wave speed relationship — a 100 Hz tone in air is 3.43 m long, which is why low-frequency noise passes through walls and around barriers that stop a 4 kHz hiss dead.
If the source or the listener is moving, the frequency shifts and the Doppler calculation applies on top of the level change; a passing train changes both pitch and loudness at once. Room and duct acoustics bring in fluid behaviour, where the transition between smooth and turbulent flow set by the Reynolds number largely determines how much broadband noise a ventilation system generates in the first place.
For regulatory work, the single-number level here is only the starting point. Occupational assessment needs an A-weighted, time-weighted average over the working day plus a peak measurement; environmental assessment usually needs LAeq over a stated period and often a statistical descriptor such as L90 for background. Underwater acoustics uses a completely different reference — 1 µPa rather than 20 µPa — so a level in water is about 26 dB higher than the same physical sound quoted the air way before you even account for the different impedance. Never compare the two directly.
This is a physics calculator, not a hearing-conservation assessment
Occupational noise exposure is assessed with a calibrated, A-weighted, time-weighted measurement over a full shift, not with a single free-field estimate. Use these numbers for design and understanding, and a type 1 or type 2 sound level meter for compliance.