Decibel Addition Formula
Learn how to add decibel levels from multiple sound sources.
Calculate combined dB using logarithmic addition with worked examples.
The Formula
Decibels cannot be added directly because the decibel scale is logarithmic, not linear. Two sound sources at 60 dB each do not produce 120 dB — they produce approximately 63 dB.
To add decibel levels correctly, you must first convert each dB value back to a linear power ratio, add those ratios, then convert the sum back to decibels. This is the fundamental principle behind noise assessment in workplaces, concert venues, and urban planning.
A useful rule of thumb: adding two identical sound levels increases the result by about 3 dB. Adding a source that is 10 dB or more below another has almost no effect on the total.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| L_total | Combined sound pressure level | dB |
| L₁, L₂, ... Lₙ | Individual sound pressure levels of each source | dB |
| log₁₀ | Base-10 logarithm | — |
Quick Reference Rules
- Two equal sources: add 3 dB (e.g., 70 + 70 = 73 dB)
- Sources differ by 1 dB: add 2.5 dB to the louder one
- Sources differ by 3 dB: add 1.8 dB to the louder one
- Sources differ by 10+ dB: the quieter source is negligible
Example 1
Two machines in a factory each produce 85 dB. What is the combined noise level?
Convert each to linear: 10^(85/10) = 10^8.5 = 3.162 × 10⁸
Add the linear values: 3.162 × 10⁸ + 3.162 × 10⁸ = 6.324 × 10⁸
Convert back: L = 10 × log₁₀(6.324 × 10⁸) = 10 × 8.801
L_total ≈ 88 dB (exactly 3 dB more than a single source, as expected)
Example 2
Three sound sources at 70 dB, 75 dB, and 80 dB are operating simultaneously. What is the total?
Convert each: 10^(70/10) = 10⁷, 10^(75/10) = 3.162 × 10⁷, 10^(80/10) = 10⁸
Sum: 10⁷ + 3.162 × 10⁷ + 10⁸ = 1.416 × 10⁸
Convert back: L = 10 × log₁₀(1.416 × 10⁸) = 10 × 8.151
L_total ≈ 81.5 dB (the 80 dB source dominates — the two quieter sources only add about 1.5 dB)
When to Use It
- Workplace noise assessments with multiple machines
- Environmental noise studies combining traffic, industrial, and ambient sources
- Audio engineering when mixing multiple sound sources
- Building acoustics — predicting noise from adjacent rooms or HVAC systems
- Regulatory compliance with noise exposure limits