The Decibel Scale
The decibel is a logarithmic unit expressing the ratio of two values (sound pressure, power, voltage). A 3 dB increase doubles acoustic power; a 10 dB increase is perceived as roughly twice as loud.
Formulas
Sound Level: L = 10 × log₁₀(I/I₀) (dB)
I₀ = 10⁻¹² W/m² (threshold of hearing)
Power ratio: dB = 10 × log₁₀(P₂/P₁)
Voltage ratio: dB = 20 × log₁₀(V₂/V₁)
Combining sources (incoherent):
L_total = 10 × log₁₀(10^(L1/10) + 10^(L2/10))Common Sound Levels (dB SPL)
- 0 dB: threshold of hearing
- 30 dB: quiet library
- 60 dB: normal conversation
- 85 dB: heavy traffic (damage threshold with prolonged exposure)
- 110 dB: concert front row
- 140 dB: jet engine at 30m (pain threshold)
Hearing Safety (NIOSH)
85 dB: 8 hours maximum
88 dB: 4 hours (every +3dB halves safe exposure time)
91 dB: 2 hours
94 dB: 1 hour
100 dB: 15 minutesConvert decibels: Free Decibel Calculator
Decibel Quick-Reference Table
| Sound level (dB SPL) | Example | Safe daily exposure |
|---|---|---|
| 0 | Threshold of hearing | No limit |
| 30 | Quiet library | No limit |
| 60 | Normal conversation | No limit |
| 85 | Heavy traffic, lawn mower | 8 hours (OSHA limit) |
| 94 | Hair dryer, power tools | 1 hour |
| 110 | Rock concert front row | 2 minutes |
| 120 | Jet engine at 30 m | Pain threshold |
| 140 | Gunshot at ear | Immediate damage |
How Decibels Work
The decibel (dB) is a logarithmic ratio: dB = 10 log₁₀(P/P₀) for power, or 20 log₁₀(A/A₀) for amplitude (pressure, voltage). For sound pressure level (SPL), P₀ = 20 μPa (threshold of human hearing). Because sound intensity spans 12 orders of magnitude (10¹²) from threshold to pain, the logarithmic scale compresses this to a 0–120 dB range.
Key rules: +3 dB doubles power (×1.41 amplitude); +6 dB doubles amplitude (×4 power); +10 dB is perceived as roughly twice as loud. Combining two identical independent sources adds +3 dB — not double the dB value. In audio electronics, 0 dBu = 0.775 V RMS; 0 dBV = 1 V RMS; 0 dBFS = full-scale digital signal.
Common Mistakes
- Adding dB levels directly: Two 80 dB sources combined = 83 dB, not 160 dB. Use the power addition formula: L_total = 10 log₁₀(10^(L₁/10) + 10^(L₂/10)).
- Forgetting the reference: dB by itself is meaningless without a reference. dB SPL uses 20 μPa; dBm uses 1 mW; dBV uses 1 V. Always specify the reference for unambiguous communication.
- Confusing the 10× and 20× formulas: Use 10 log₁₀ for power quantities (W, W/m²); use 20 log₁₀ for amplitude quantities (Pa, V). Using 10 for pressure gives half the correct dB value.
Frequently Asked Questions
Human perception of loudness and brightness is approximately logarithmic (Weber-Fechner Law). Each tenfold increase in sound intensity is perceived as roughly twice as loud. A linear scale would require numbers from 1 to 1,000,000,000,000 to describe audible sounds — unwieldy compared to 0–120 dB.
dB SPL measures physical sound pressure equally at all frequencies. dBA applies a weighting curve (the A-weighting filter) that mimics human hearing sensitivity — boosting 1–6 kHz (where hearing is most sensitive) and attenuating bass and high frequencies. Occupational noise limits (85 dBA, 8 hours) use dBA because low-frequency noise at 85 dB SPL is less damaging to hearing than high-frequency noise at the same level.
Typical STC (Sound Transmission Class) values: single-pane window 27 dB; standard drywall (single layer each side) 33–36 dB; well-built partition 45–50 dB; concrete block wall 50–55 dB. A 40 dB reduction means sound power reaching the other side is 1/10,000 of the source — enough to reduce a 90 dB construction site noise to approximately 50 dB inside a well-constructed office.