Formula & Calculator
Noise Level Attenuation with Distance
Estimates how much sound pressure level decreases as distance from a point noise source increases (inverse square law for sound).
Interpretation
L2 = L1 − 20×log₁₀(d2/d1). Sound level decrease with distance (inverse square law). Used to predict noise impact from point sources and to design noise barriers.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| L2 | Sound level at new distance | dB |
| L1 | Sound level at reference distance | dB |
| d2 | New distance from source | m |
| d1 | Reference distance from source | m |
What it means
This formula describes the geometric spreading of sound waves from a point source. The sound pressure level (L) decreases by 20 dB per decade of distance increase in a free‑field condition. It is derived from the inverse square law for sound intensity. This is used in environmental noise assessment to estimate noise levels at receptors (e.g., residences) from industrial sources, traffic, or construction. It does not account for atmospheric absorption or ground effects, so it provides a conservative estimate. Example: A compressor produces 90 dB at 1 m. The level at 10 m is L2 = 90 − 20×log₁₀(10/1) = 90 − 20 = 70 dB. At 100 m, L2 = 90 − 40 = 50 dB. This helps in determining setback distances for noise‑sensitive areas.
Worked example
Noise Attenuation with Distance – Two Examples
Real‑World| Parameter | Value |
|---|---|
| L1 | 90 dB |
| d1 | 1 m |
| d2 | 10 m |
| Parameter | Value |
|---|---|
| L1 | 95 dB |
| d1 | 10 m |
| d2 | 100 m |
Common mistakes
- Distance ratio: d2/d1 must be greater than 1 for attenuation; if d2 < d1, the level increases (i.e., moving closer).
- Units: d1 and d2 in the same units (e.g., metres).
- Inverse square law: This formula assumes a point source in free field (no reflections). For line sources or indoor spaces, use different models.
- Atmospheric absorption: Not included – for long distances, air absorption may be significant (add an extra term).
- Background noise: The formula gives the level from the source alone; combine with background using logarithmic addition.
Applications
Noise level attenuation with distance follows the inverse square law for a point source, L₂ = L₁ − 20·log₁₀(d₂/d₁). This equation describes the reduction in sound pressure level as distance from the source increases. Environmental acousticians use it to assess noise impacts from transportation, industrial, and construction activities, and to design noise barriers or buffer zones. It is also applied in occupational health to determine safe distances from noisy equipment. By calculating attenuation, engineers can predict noise levels at receivers, evaluate compliance with local noise ordinances, and implement mitigation measures such as silencers, enclosures, or setback distances.
- Noise impact assessment for highways, railways, and airports
- Industrial noise control and compliance with occupational limits
- Design of sound barriers and noise‑reducing enclosures
- Urban planning and zoning to minimise noise exposure
- Community complaint investigation and mitigation design
Frequently Asked Questions
For a point source in free space, the sound pressure level decreases by 6 dB per doubling of distance (inverse square law). The formula is L2 = L1 – 20·log10(d2/d1), where L1 is the level at distance d1, and L2 is the level at distance d2.
Applying the point‑source inverse‑square formula to line sources (like highways) or in environments with significant reflections, where a different attenuation rate applies. For line sources, the level drops by 3 dB per doubling of distance (cylindrical spreading).
- Point source (spherical spreading): 6 dB per doubling of distance.
- Line source (cylindrical spreading): 3 dB per doubling of distance.
- Infinite plane source: no attenuation with distance.
L2 = 80 – 20·log10(50/10) = 80 – 20·log10(5) = 80 – 20×0.699 = 80 – 13.98 ≈ 66 dB.
- Atmospheric absorption (depends on frequency, temperature, humidity).
- Ground reflection and absorption.
- Obstructions (buildings, barriers).
- Wind and temperature gradients (refraction).
Barriers (walls, berms) can reduce noise by diffraction. The attenuation depends on the path difference between the direct and diffracted paths. Simple empirical formulas or modelling software are used for accurate prediction.
Sound pressure level (Lp) is what we measure (dB) and depends on distance. Sound power level (Lw) is the total acoustic energy emitted by the source and is independent of distance. The formula relates Lp to Lw and distance.
The factor 20 comes from the definition of decibels for pressure ratios. Since sound pressure is inversely proportional to distance (for a point source), the level difference is 20·log10(d2/d1).
Rearrange the formula: d2 = d1 × 10^((L1 – L2)/20). For example, if L1=90 dB at 1 m, and you want L2=70 dB, d2 = 1 × 10^((90‑70)/20) = 1 × 10^1 = 10 m.
It assumes free‑field propagation with no reflections, no absorption, and no barriers. In real outdoor environments, more complex models (e.g., ISO 9613) are used. The formula is a good approximation for open areas at moderate distances.