Decibel Sound Level Formula
Sound intensity in decibels: L = 10 × log₁₀(I/I₀).
How loudness scales with intensity.
Why +10 dB sounds twice as loud but represents 10× more energy.
The Formula
The decibel (dB) scale measures sound intensity on a logarithmic scale. I is the measured intensity (W/m²) and I₀ is the reference intensity — the threshold of human hearing at 10⁻¹² W/m².
The logarithmic scale matches human perception: we perceive equal ratios of intensity as equal steps of loudness. Every 10 dB increase represents 10× more energy but sounds roughly twice as loud to the human ear. Every 3 dB increase represents approximately 2× more energy (a just-noticeable difference in loudness).
For sound pressure (more common in audio engineering): L_p = 20 × log₁₀(p / p₀), where p₀ = 20 μPa (threshold of hearing in air).
Reference sound levels: Threshold of hearing: 0 dB. Whisper: 30 dB. Normal conversation: 60 dB. Heavy traffic: 85 dB. Concert: 110 dB. Jet engine at 30 m: 130 dB. Threshold of pain: ~120–140 dB. Damage to hearing begins with sustained exposure above 85 dB (8 hours) or brief exposure above 120 dB.
In audio production, 0 dBFS (full scale) is the maximum digital level. Levels are expressed as negative numbers: −6 dBFS is 6 dB below maximum, −18 dBFS is a common recording level for headroom.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| L | Sound level | decibels (dB) |
| I | Sound intensity | W/m² |
| I₀ | Reference intensity (hearing threshold) | 10⁻¹² W/m² |
Example 1
A sound has intensity 10⁻⁶ W/m². What is its level in dB?
L = 10 × log₁₀(10⁻⁶ / 10⁻¹²) = 10 × log₁₀(10⁶) = 10 × 6
L = 60 dB (normal conversation level)
Example 2
Two identical speakers both at 90 dB — what is the combined level?
Adding equal sources: L_combined = L + 10 × log₁₀(2) = 90 + 3
L = 93 dB (doubling sources adds only 3 dB)
When to Use It
- Audio engineering: mixing, mastering, gain staging
- Hearing protection: calculating safe exposure times
- Acoustic design of studios, concert halls, and classrooms
- Speaker and amplifier power calculations