Sound levels are logarithmic—they cannot be added directly. Two identical sources produce +3 dB, not double the dB value. Enter each source’s SPL below to find the true combined level.
What Is a Decibel?
The decibel (dB) is a logarithmic unit that expresses the ratio between two quantities. It was introduced because human sensory systems—hearing and vision alike—respond logarithmically to stimulus intensity. A linear doubling of sound power corresponds to an increase of 3 dB; a perceived doubling of loudness requires approximately 10 dB. This compression allows us to describe the entire range of human hearing, from the faintest detectable whisper to the threshold of pain, using a manageable scale of 0 to 130 rather than a factor of 10,000,000,000,000 in pressure.
In acoustics, the standard reference is dB SPL (sound pressure level), defined relative to 20 μPa—the nominal threshold of human hearing at 1 kHz. A measurement of 0 dB SPL does not mean silence; it means the pressure fluctuation equals the reference threshold.
How to Add Decibel Values
Because decibels are logarithmic, adding two SPL measurements requires converting each to linear power, summing, and converting back:
Ltotal = 10 × log10(Σ 10Li / 10)
This produces several counter-intuitive results that are essential to internalise:
- Two identical sources: +3 dB (e.g., 80 + 80 = 83 dB).
- Ten identical sources: +10 dB (e.g., ten 80 dB sources = 90 dB).
- Adding a source 10 dB quieter than the dominant source: +0.4 dB—functionally negligible.
- Adding a source 3 dB quieter: +1.8 dB.
The key insight: in any combination, the loudest source dominates. Quieter sources contribute progressively less as the gap increases. This explains why, in a factory with one machine at 95 dB and twenty machines at 80 dB, the measured level is still approximately 95 dB.
The Inverse Square Law for Sound
Sound radiating from a point source in free space (no reflections) spreads over an ever-increasing spherical area. Since the area of a sphere grows with the square of its radius, sound intensity—power per unit area—decreases with the square of the distance. In decibels, this translates to:
ΔL = 20 × log10(d1 / d2)
Doubling the distance from a source reduces SPL by 6.02 dB. Halving the distance increases it by the same amount. This law governs speaker placement, PA system coverage, noise barrier design, and the calculation of safe listening distances.
Common Sound Levels for Reference
| Source | Typical SPL | Exposure notes |
|---|---|---|
| Threshold of hearing | 0 dB | Reference level (20 μPa) |
| Quiet library | 30 dB | Background ambient |
| Normal conversation (1 m) | 60 dB | Comfortable for extended periods |
| Vacuum cleaner | 75 dB | Below damage threshold |
| Heavy traffic | 85 dB | Damage threshold (8 h/day) |
| Loud headphones | 100 dB | Safe limit: 15 min/day |
| Rock concert (front row) | 110 dB | Hearing protection recommended |
| Threshold of pain | 130 dB | Immediate damage risk |
Hearing Safety and Noise Exposure
The relationship between SPL and safe exposure time is exponential. NIOSH guidelines indicate that for every 3 dB increase above 85 dB, the permissible exposure time halves: 85 dB allows 8 hours, 88 dB allows 4 hours, 91 dB allows 2 hours, and so on. At 100 dB, the safe limit is approximately 15 minutes. Musicians, sound engineers, and concert-goers routinely exceed these limits, making noise-induced hearing loss the most common occupational hazard in the music industry.