Noise Level Attenuation Calculator
Predict how sound level falls with distance from a source using L₂ = L₁ − 20·log₁₀(d₂/d₁) for point sources (6 dB per doubling) or the 10·log₁₀ line-source variant (3 dB per doubling) — with per-distance readouts, a live propagation schematic, and companion noise-mapping metrics.
Geometric spreading is the dominant attenuation mechanism
Sound from a point source radiates in all directions, and its energy is spread over the surface of an expanding sphere. The surface area of a sphere grows with the square of its radius, so the intensity (power per unit area) falls off as 1/d². Since sound level is proportional to the logarithm of intensity, the level falls off as 20·log₁₀(d) — the famous 6 dB per doubling of distance. This is a pure geometric effect and does not depend on frequency, air temperature or humidity.
Line sources attenuate half as fast
A long highway, railway or pipeline behaves acoustically as a line rather than a point source at distances less than its length. Its energy spreads over a cylindrical surface, which grows linearly with distance rather than as the square, so the intensity falls off as 1/d and the level falls off as 10·log₁₀(d) — only 3 dB per doubling of distance. This is why noise from a busy motorway is audible over much greater distances than noise from a single truck, even though the levels close to each are similar.
Real environments add extra attenuation
The 20·log₁₀ formula is a free-field ideal. In practice, atmospheric absorption (especially at high frequencies and low humidity), ground effect (interference from reflected sound), barriers, vegetation and wind gradients all reduce levels further. On a clear, dry day at 2 kHz, molecular absorption can add 0.01–0.02 dB/m — that is 10–20 dB over 1 km on top of the geometric loss. This is why measured community noise is typically 3–10 dB below the free-field prediction, especially at long range.
Distance is the first and cheapest control
Before considering barriers, silencers or enclosures, the acoustic designer asks whether the receiver can simply be moved further from the source. Because level drops by 6 dB per doubling of distance for a point source, moving a receiver from 10 m to 40 m (four doublings, or 2 factor-of-4 steps... precisely 20·log₁₀(4) = 12 dB) is often cheaper than any hardware solution. Setback distances, buffer zones, and thoughtful siting of noise-sensitive buildings are the most cost-effective noise control measures available.
How to Use This Calculator
The distance-attenuation relation links four quantities: L₁, L₂, d₁ and d₂. Pick which one you want the tool to solve, then fill in the others.
1. Choose what to solve for
Pick L₂, L₁ or d₂ from the toggle. The calculator rearranges the propagation formula automatically for whichever variable you choose — L₂ is the usual target for forward prediction, L₁ when working back from a limit, and d₂ when finding how far to move a receiver to hit a target level.
2. Pick the propagation model
Point source (20·log) for single machines, isolated vehicles, exhaust stacks and most discrete equipment. Line source (10·log) for highways, railways, long pipe runs, and any extended source at close range. The toggle updates the formula and every downstream metric.
3. Keep units consistent
d₁ and d₂ must be in the same unit — the formula uses only their ratio, so mixing metres with kilometres is the classic error. The presets give common distances in metres, but the calculator works for feet, miles, or any other unit as long as both fields agree.
4. Read the result and verdict
The result card shows the solved level in dB, plus the attenuation ΔL and the doubling-equivalent distance ratio. A colour-coded verdict classifies the level from very quiet (WHO night limit) through quiet, moderate, loud, dangerous and hazardous, benchmarked against typical environmental and occupational thresholds.
5. Check the schematic and charts
The schematic shows expanding spherical wavefronts from the source, with d₁ and d₂ marked, the levels at each distance labelled, and the attenuation and doubling rule shown alongside. The main chart plots L₂ against d₂ for three reference levels.
6. Switch to Advanced for depth
Advanced mode adds a level-vs-reference chart comparing point and line models, a sensitivity analysis, and companion metrics including the linear pressure ratio, the acoustic power ratio, the level at 2×, 10× and 100× the reference distance, and compliance checks against WHO, EU and OSHA limits.
What This Tool Offers
Built around the free-field propagation relation L₂ = L₁ − 20·log₁₀(d₂/d₁), the fundamental link between source level, distance and received level.
Three-way solving
Solve for the level at the target distance, the reference level at d₁, or the distance required to reach a target level. The calculator rearranges the formula for whichever variable you choose.
Point and line source models
Toggle between the 20·log₁₀ point-source model (6 dB per doubling) and the 10·log₁₀ line-source model (3 dB per doubling) — both are used throughout noise assessment.
Live propagation schematic
Expanding spherical wavefronts from the source, with the reference and target distances marked, their levels labelled, and the attenuation and doubling rule shown in the centre of the diagram.
Sound-class verdict
Each result is classified from very quiet (WHO night limit) through quiet, moderate, loud, dangerous and hazardous, with a coloured badge and reference class.
Companion metrics
Pressure ratio, acoustic power ratio, level at 2×/10×/100× the reference distance, doubling count, WHO/EU/OSHA compliance flags, and safe exposure time — all live.
Reference tables
Attenuation vs. distance ratio for both models, typical source levels, common regulatory limits, and a full variable reference with unit notes.
Reference Tables & Conversions
A quick three-field attenuation solver, a distance-ratio lookup table, and typical regulatory limits.
Quick distance-attenuation solver — enter any two, read the third
Attenuation Lookup — Distance Ratio vs. ΔL
Computed from ΔL = 20·log₁₀(r) for point sources (left column) and ΔL = 10·log₁₀(r) for line sources (right column).
| Distance ratio (d₂/d₁) | ΔL point (20·log) | ΔL line (10·log) | Doublings | Pressure ratio p₂/p₁ | Power ratio |
|---|
Typical Source Levels and Regulatory Limits
Representative A-weighted sound pressure levels and common regulatory limits for context.
| Source / limit | Level (dB) | Category | Notes |
|---|---|---|---|
| WHO night-time guideline (residential) | 40 – 45 | Regulatory limit | Outdoor level outside bedroom windows |
| WHO day/evening guideline | 50 – 55 | Regulatory limit | Outdoor residential, day/evening |
| EU Environmental Noise Directive (Lden) | 55 – 60 | Regulatory limit | Member state-specific values |
| UK night-time limit (BS 4142) | Background + 10 | Assessment method | Rating level above background |
| US OSHA 8-h workplace limit | 85 (TWA) | Regulatory limit | Hearing protection required above this |
| NIOSH recommended exposure limit | 85 (TWA, 3 dB exchange) | Recommended limit | More protective than OSHA |
| Domestic refrigerator at 1 m | 35 – 45 | Typical source | Barely audible indoors |
| Normal conversation at 1 m | 55 – 65 | Typical source | Speech clearly intelligible |
| Passenger car at 10 m (highway) | 65 – 75 | Typical source | Single vehicle, moderate speed |
| Lawnmower at 1 m | 85 – 90 | Typical source | Hearing-protection range |
| Heavy truck at 10 m | 80 – 90 | Typical source | Dominant highway noise contributor |
| Rock concert / nightclub (audience) | 95 – 110 | Typical source | Exposure limits in minutes |
| Handheld drill / chainsaw at 1 m | 100 – 110 | Typical source | Hearing protection mandatory |
| Jet engine at 25 m | 130 – 140 | Typical source | Immediate damage risk |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| L₁ | Reference sound level at distance d₁ | 0 – 150 | dB SPL |
| L₂ | Level at target distance d₂ | 0 – 150 | dB SPL |
| d₁ | Reference distance | 0.1 – 10⁴ | m (or any consistent unit) |
| d₂ | Target distance | 0.1 – 10⁵ | m (same unit as d₁) |
| r = d₂/d₁ | Distance ratio | 0.01 – 10⁵ | — |
| ΔL | Attenuation (positive for loss) | −20 to 100 | dB |
| p₂/p₁ | Pressure ratio | 0.01 – 1 | — |
| W₂/W₁ | Acoustic power ratio (per unit area) | 10⁻⁴ – 1 | — |
What Does L₂ = L₁ − 20·log₁₀(d₂/d₁) Measure?
The distance-attenuation formula gives the sound level at any distance from a point source, given its level at a reference distance. It is the oldest and most fundamental tool in acoustics — it captures the geometric spreading of sound as it radiates from a compact source into free space, and it explains why a whisper carries only a metre or two while a jet engine can be heard kilometres away. The formula is a direct consequence of the inverse-square law: intensity falls off as 1/d², and the logarithm of that ratio is 20·log₁₀(d₂/d₁).
The "point source" assumption is that the source is small compared to the distances involved, so that sound radiates equally in all directions. A machine on a factory floor, a vehicle, a loudspeaker, an exhaust stack — each is a point source for receivers more than a few source dimensions away. Long linear sources (highways, railways, pipelines, chains of cooling towers) behave differently, following the cylindrical 10·log₁₀ rule instead.
Noise Attenuation Formula
The defining relationship for a point source is:
For a line source (highway, railway, long pipe at close range):
Rearranged to solve for the other two unknowns (point source):
d₂ = d₁ · 10(L₁ − L₂) / 20
The doubling rule — the single most useful rule of thumb in acoustics:
line source: −3.01 dB per doubling of distance
Converting level difference to physical ratios:
intensity ratio = 10−ΔL/10
Propagation Variables
| Symbol | Meaning | Notes |
|---|---|---|
| L₁ | Level at the reference distance | Usually measured at 1 m or provided by the manufacturer |
| L₂ | Level at the target distance | Computed from L₁ and the ratio d₂/d₁ |
| d₁ | Reference distance | Must be in the same units as d₂ |
| d₂ | Target distance | Must be in the same units as d₁ |
| r | Distance ratio d₂/d₁ | Dimensionless; the only distance input the formula requires |
| ΔL | Attenuation | L₁ − L₂; positive when L₂ is lower |
| p | Sound pressure | RMS fluctuation above/below atmospheric pressure |
| I | Sound intensity | Acoustic power per unit area, falling off as 1/d² |
How Each Variable Affects the Result
The relation is linear in L₁ and logarithmic in the distance ratio. Increasing L₁ by 1 dB increases L₂ by exactly 1 dB — the level simply shifts. Doubling the distance ratio reduces the level by 6.02 dB for a point source (or 3.01 dB for a line source) regardless of the absolute distance: moving from 1 m to 2 m has the same effect as moving from 100 m to 200 m.
This is why decibel arithmetic is so counterintuitive. If you want to halve the perceived loudness, you need a 10 dB reduction, which corresponds to a distance ratio of 10^(10/20) = 3.16 for a point source. If you want a 20 dB reduction — a factor of 100 in acoustic intensity — you need a distance ratio of 10, so from 10 m to 100 m, or from 100 m to 1 km. And because real propagation adds atmospheric and ground attenuation on top of the geometric loss, the actual distance required is often less than the free-field formula predicts, especially at frequencies above 1 kHz.
Line sources behave very differently. A highway, for example, only attenuates at 3 dB per doubling of distance, so it remains audible much further away than a single car. This is why noise contours around motorways, railways and industrial plants with long rows of equipment spread out to kilometres rather than hundreds of metres, and why setback distances for new residential development near such corridors must be far larger than for a single point source of the same nominal level.
Attenuation Calculation Examples
L₂ = 90 − 20·log₁₀(100/1) = 90 − 40 = 50.0 dB
L₂ = 90 − 10·log₁₀(100/1) = 90 − 20 = 70.0 dB
L₁ = 55 + 20·log₁₀(200/1) = 55 + 46.02 = 101.0 dB
d₂/d₁ = 10^((90−45)/20) = 10^2.25 = 177.8 × d₁ (e.g., 178 m from a 1 m reference)
Frequently Asked Questions
Because the acoustic power from a point source spreads over an ever-larger spherical surface. The area of a sphere is 4πr², so at twice the distance the same power is spread over four times the area, and the intensity (power per unit area) is one quarter. A factor of 4 in intensity is 10·log₁₀(4) = 6.02 dB. This is the inverse-square law, and it is the single most reliable propagation rule in acoustics. It holds for any point source in free space, at any frequency, independent of air temperature, humidity and wind — those factors come in only as additional attenuation on top.
When the source is long compared to the distance to the receiver — typically when the receiver is closer than about half the length of the source. A single car is a point source; a 1 km queue of traffic at 100 m is effectively a line. A single rooftop fan is a point source; a row of 20 cooling towers along a building is effectively a line. Railways and highways are line sources for distances up to a few hundred metres. Beyond the point where the source length becomes small compared to the distance, the line source transitions to a point source — a busy motorway at 5 km is heard as a single distant source, not an infinite line.
No — it only accounts for geometric spreading in an ideal free field. Real attenuation from a barrier is typically 5–20 dB depending on the barrier's height and the geometry of the source-barrier-receiver path. Dense vegetation adds 1–3 dB per 10 m of depth. Buildings add 5–15 dB per row of houses, depending on their height and spacing. Ground effect (reflection vs. absorption) adds ±3 dB depending on surface type. Atmospheric absorption adds 0.005–0.02 dB/m at mid frequencies and much more at high frequencies. In practice, real community noise is 3–15 dB below the free-field prediction because all these effects stack up.
dB SPL is the unweighted sound pressure level, treating all audible frequencies equally. dB(A) applies the A-weighting filter, which mimics the ear's reduced sensitivity at low and high frequencies — it is the standard for environmental and occupational noise regulation. dB(C) applies a flatter weighting that emphasises low frequencies more, and is used for assessing peak levels or low-frequency noise. All three obey the same 20·log₁₀ distance-attenuation rule, so the propagation calculation is unchanged — but the numerical result must be expressed in the same weighting as the input. If L₁ is in dB(A), L₂ will be in dB(A).
Rearrange the formula: d₂/d₁ = 10^((L₁ − L₂)/20) for a point source. For example, if a compressor produces 100 dB at 1 m and your local night-time limit is 45 dB, you need d₂/d₁ = 10^(55/20) = 10^2.75 = 562, so a setback of 562 m. If the source is a line (a long row of compressors), the required distance is only 10^(55/10) = 316,000 — but wait, the line-source formula grows much more slowly, so in practice line sources require less distance than the point-source formula suggests. Real-world predictions always add empirical corrections for ground effect and atmospheric absorption, which typically reduce the required distance by 10–30 %.
Yes, but only through the atmospheric absorption term, not the geometric term. Molecular absorption by oxygen and nitrogen is strongly frequency-dependent and reaches a minimum at low humidity — around 20–40 % RH — for frequencies above 1 kHz. At 4 kHz and 20 % RH, absorption is about 0.03 dB/m, whereas at 70 % RH it drops to about 0.015 dB/m. Over a kilometre that is a 15 dB difference in the high-frequency content, which changes the perceived quality of the noise (hiss vs rumble) but has a modest effect on the overall A-weighted level. For road traffic and industrial noise, atmospheric absorption typically adds 1–5 dB to the geometric loss over 1 km.
Because the highway is a line source, and its sound adds coherently along its length. Each individual car is a point source attenuating at 6 dB per doubling, but a long row of cars — each contributing incoherently at the receiver — behaves like a continuous line source attenuating at only 3 dB per doubling. The effect is dramatic: at 500 m, a single car might be 30 dB below its 10 m level (in silence), whereas a continuous stream of traffic along 1 km of road would be only 15 dB below. This is also why wind turbine noise and railway noise propagate so far compared to isolated machines of similar source level.
Not directly. Underwater acoustics uses a different reference pressure (1 µPa instead of 20 µPa) and different units (dB re 1 µPa). The geometric spreading formula is the same in form — 20·log₁₀ for spherical spreading and 10·log₁₀ for cylindrical spreading — but underwater propagation is strongly affected by the sound-speed profile (which refracts sound), by absorption (which depends on frequency and is much higher than in air), and by the sea surface and bottom boundaries. Underwater noise from ships, sonar and pile-driving is a specialised field with its own propagation models (e.g., RAM, Bellhop).
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
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