Sound Intensity Level (Decibels)
Calculate sound intensity level in decibels using L = 10 log₁₀(I/I₀), where I₀ is the threshold of human hearing.
The Formula
Sound intensity level measures how loud a sound is on a logarithmic scale. The unit is the decibel (dB).
Because human hearing spans an enormous range of intensities (a factor of about 10¹²), a logarithmic scale is far more practical than a linear one. Every increase of 10 dB represents a tenfold increase in sound intensity.
Variables
| Symbol | Meaning |
|---|---|
| L | Sound intensity level (in decibels, dB) |
| I | Sound intensity (in watts per square meter, W/m²) |
| I₀ | Reference intensity — threshold of hearing (1 × 10⁻¹² W/m²) |
Common Sound Levels
| Source | Approximate Level (dB) |
|---|---|
| Threshold of hearing | 0 dB |
| Whisper | 20 dB |
| Normal conversation | 60 dB |
| Vacuum cleaner | 70 dB |
| Rock concert | 110 dB |
| Jet engine at 30 m | 140 dB |
Example 1
A sound has an intensity of 1 × 10⁻⁵ W/m². What is the sound level in decibels?
L = 10 log₁₀(I / I₀)
L = 10 log₁₀(1 × 10⁻⁵ / 1 × 10⁻¹²)
L = 10 log₁₀(10⁷)
L = 10 × 7
L = 70 dB (similar to a vacuum cleaner)
Example 2
Two sound sources each produce 80 dB. What is the combined level?
First, convert each to intensity: 80 = 10 log₁₀(I / 10⁻¹²)
I = 10⁻¹² × 10⁸ = 1 × 10⁻⁴ W/m² for each source
Total intensity: I_total = 2 × 10⁻⁴ W/m²
L = 10 log₁₀(2 × 10⁻⁴ / 10⁻¹²) = 10 log₁₀(2 × 10⁸)
L = 10 × (8 + log₁₀(2)) = 10 × (8 + 0.301)
L ≈ 83 dB (doubling intensity adds about 3 dB, not double the decibels)
When to Use It
Use this formula when working with sound measurements and noise levels.
- Environmental noise assessment and regulations
- Audio engineering and speaker design
- Workplace safety (hearing protection requirements)
- Acoustic design of buildings and concert halls
- Comparing relative loudness of different sources