Sound Intensity Formula
Reference for I = P / (4πr²) and the inverse square law.
Calculate how sound spreads with distance in dB with examples for speakers and noise levels.
Need to calculate, not just reference? Use the interactive version.
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The Formula
I = P / (4πr²)
Sound intensity is the power per unit area at a given distance from a source. Sound follows the inverse square law — doubling the distance reduces intensity to one quarter.
Variables
| Symbol | Meaning |
|---|---|
| I | Sound intensity (W/m²) |
| P | Sound power of the source (watts) |
| r | Distance from the source (meters) |
| 4πr² | Surface area of a sphere at distance r |
Example 1
A speaker outputs 0.5 W of sound power. Find the intensity at 3 meters.
I = 0.5 / (4π × 3²) = 0.5 / (4π × 9)
I = 0.5 / 113.1
I ≈ 0.00442 W/m² ≈ 4.42 × 10⁻³ W/m²
Example 2
At 2 m from a source, intensity is 0.01 W/m². What is it at 6 m?
By the inverse square law: I₂/I₁ = (r₁/r₂)²
I₂ = 0.01 × (2/6)² = 0.01 × (1/3)² = 0.01 × 1/9
I₂ ≈ 0.00111 W/m² (one-ninth the original intensity)
When to Use It
Use the sound intensity formula when:
- Calculating how loud a source will be at a given distance
- Designing speaker placement and sound systems
- Assessing noise levels for workplace safety compliance
- Understanding how sound diminishes with distance
Key Notes
- The inverse square law assumes a point source radiating uniformly in free space with no reflections — indoors or near reflective surfaces, sound decays more slowly than predicted; directional speakers also deviate significantly from this model
- Doubling the distance drops intensity to 1/4, which equals a 6 dB reduction — this "6 dB per doubling" rule is used by sound engineers for speaker placement and occupational noise assessments
- Converting intensity to decibels: L = 10 × log₁₀(I / I₀), where the reference I₀ = 10⁻¹² W/m² (threshold of hearing) — a 10 dB increase multiplies intensity by 10, so 80 dB is 100× more intense than 60 dB, not 1.33× more
- Sound power P (watts) is a fixed property of the source; intensity I (W/m²) depends on your distance from it — a 0.1 W speaker may be loud at 0.5 m but inaudible at 50 m; always distinguish between power and intensity when reading specifications