Sound Intensity Level Formula
The sound intensity level formula L = 10 log(I/I0) converts sound intensity to decibels.
Learn with worked examples.
The Formula
The sound intensity level formula converts the physical intensity of sound (measured in watts per square meter) into decibels (dB). The decibel scale is logarithmic because the human ear perceives loudness logarithmically over an enormous range.
The reference intensity I₀ is the threshold of human hearing, defined as 1 × 10⁻¹² W/m². This means 0 dB corresponds to the faintest sound a healthy young person can hear. Normal conversation is about 60 dB, and the threshold of pain is around 120-130 dB.
Because the scale is logarithmic, every increase of 10 dB represents a tenfold increase in intensity. However, our perception of loudness roughly doubles for every 10 dB increase. Two sound sources of equal intensity together produce only a 3 dB increase, not double the decibel value.
Variables
| Symbol | Meaning |
|---|---|
| L | Sound intensity level (decibels, dB) |
| I | Sound intensity (watts per square meter, W/m²) |
| I₀ | Reference intensity (1 × 10⁻¹² W/m²) |
Example 1
A sound has an intensity of 1 × 10⁻⁶ W/m². What is the sound level in decibels?
Apply the formula: L = 10 × log₁₀(1 × 10⁻⁶ / 1 × 10⁻¹²)
L = 10 × log₁₀(1 × 10⁶)
L = 10 × 6
L = 60 dB (about the level of normal conversation)
Example 2
A rock concert produces 110 dB. What is the actual sound intensity?
Rearrange: I = I₀ × 10^(L/10) = 1 × 10⁻¹² × 10^(110/10)
I = 10⁻¹² × 10¹¹
I = 0.1 W/m² (100 billion times the threshold of hearing)
When to Use It
Use the sound intensity level formula to convert between physical sound intensity and the decibel scale.
- Measuring noise levels for workplace safety compliance
- Designing soundproofing and acoustic treatment
- Calculating how sound levels change with distance from a source
- Comparing the loudness of different sound sources