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Inverse Square Law Calculator

Calculate how sound pressure level drops with distance. Based on the inverse square law — every doubling of distance reduces SPL by 6 dB in free field conditions.

Presets:

Source & Distance

Source SPL at Reference Distance

dB

Reference Distance (d₁)

m

1 m100 m

Target Distance (d₂)

m

1 m100 m

Results

SPL at Target Distance

74.0

dB

dB Change

−20.0 dB

Distance Ratio

10 : 1

Intensity Ratio

1 : 100

Intensity Factor

0.01×

Formula Applied

SPL₂ = SPL₁ − 20 × log₁₀(d₂ / d₁)

= 94 − 20 × log₁₀(10/1) = 74.0 dB

Practical dB vs Distance Reference

Distance Ratio (d₂/d₁)	dB Change	Intensity Change	Example
0.5× (half)	+6 dB	4× more intense	Move speaker twice as close
1× (same)	0 dB	No change	Reference point
2× (double)	−6 dB	4× less intense	6 dB rule of thumb
3×	−9.5 dB	9× less intense	Triple the distance
10×	−20 dB	100× less intense	1 m → 10 m
100×	−40 dB	10,000× less intense	1 m → 100 m

Practical Applications

Noise Ordinance Compliance — If a speaker measures 110 dB at 1 m, enforcement points at 30 m would receive 110 − 20×log₁₀(30) = 80.5 dB, potentially exceeding local limits.
Speaker Coverage Design — PA engineers use the inverse square law to calculate throw distance from line arrays and delay stacks, ensuring even coverage throughout a venue.
Hearing Protection — OSHA limits require protection above 85 dB. Workers can calculate the safe stand-off distance from machinery at a known SPL rating.
Microphone Placement — Recording engineers use the 3:1 rule and inverse square law to minimize bleed between microphones at different distances.
Outdoor Acoustics — Community noise assessments predict residential exposure from industrial sources kilometers away using this foundational relationship.

Understanding the Inverse Square Law

In a free, unobstructed sound field, sound radiates outward from a point source as an expanding sphere. The surface area of a sphere grows as 4πr². Because the total acoustic power is conserved, sound intensity (power per unit area, measured in W/m²) decreases proportionally to 1/r². This is why the relationship is called an inverse square law — intensity is inversely proportional to the square of the distance. This makes it the essential sound intensity calculator relationship for any outdoor or free-field scenario where you need to predict noise level at distance.

The decibel scale is logarithmic: a 10× reduction in intensity equals −10 dB; a 100× reduction equals −20 dB. Since intensity ∝ 1/d², a doubling of distance causes a 4× drop in intensity, which equals −6 dB (−10 × log₁₀(4) ≈ −6.02 dB). This “6 dB per doubling” rule — sometimes called the decibel-distance rule — is one of the most-cited principles in acoustics and audio engineering. To convert between intensity, SPL, and sound pressure in pascals, use the SPL calculator.

Important limitations: The inverse square law applies strictly to a point source in a free field with no reflections or obstructions. In real rooms, reflected sound adds a reverberant field that makes the actual level drop slower than the free-field prediction beyond the critical distance (sometimes called the “reverberation radius”). Line sources such as long straight traffic roads or extended speaker arrays follow a cylindrical spreading law of −3 dB per doubling, not −6 dB. Atmospheric absorption adds additional high-frequency losses over very long distances that this formula does not account for. For indoor predictions with room reflections, room acoustics software or the room mode calculator gives more realistic estimates.

How do you calculate SPL at a new distance? Step-by-step example

The tool uses one formula throughout: SPL₂ = SPL₁ − 20 × log₁₀(d₂ / d₁), where SPL₁ is the known level at reference distance d₁, and SPL₂ is the predicted level at target distance d₂. The −20 dB coefficient comes directly from the inverse square law: intensity falls as 1/r², so the pressure-domain factor of 20 (not 10) applies when working in SPL units.

Concrete example: A loudspeaker measures 100 dB at 1 m. What is the SPL at 25 m?

Compute the distance ratio: 25 / 1 = 25
Take the log: log₁₀(25) = 1.398
Scale by 20: 20 × 1.398 = 27.96 dB drop
Subtract: 100 − 27.96 = 72.0 dB at 25 m

Two common reference values worth memorising: a doubling of distance always yields −6 dB (distance ratio 2, log₁₀(2) × 20 = 6.02 dB), and a 10× increase in distance yields −20 dB. For multi-source environments where you need to add SPL contributions rather than track a single source across distance, see the decibel calculator. To assess whether the resulting level poses a hearing risk, the noise exposure calculator converts SPL and duration into a dose percentage against NIOSH/OSHA limits.

Frequently Asked Questions

Why does sound level drop 6 dB when distance doubles?

Sound radiates as a sphere, so its energy spreads over an area proportional to distance squared. Doubling the distance quadruples the area, spreading the same energy 4× thinner — a 4× reduction in intensity equals 10×log₁₀(4) ≈ 6 dB. This is the most fundamental rule in outdoor acoustics and PA system design.

Does the inverse square law work indoors?

Only in the "direct field" close to the source. Beyond the critical distance (where reflected energy equals direct energy), sound levels drop much more slowly — less than 6 dB per doubling. In a highly reverberant room, levels can be nearly constant everywhere. The inverse square law is most accurate outdoors or in anechoic conditions.

What is the inverse square law used for in audio engineering?

It is used to predict speaker throw distance, calculate safe noise exposure distances, design microphone arrays, set up delay towers in live sound, and ensure PA coverage uniformity. It also underpins noise barrier design, industrial hygiene assessments, and environmental impact studies for airports and highways.

Does the 6 dB rule apply to line sources like road traffic noise?

No. The 6 dB per distance-doubling rule applies to point sources (spherical spreading). A long, straight line source — such as a busy road or railway — spreads sound in a cylinder rather than a sphere, so levels drop by only 3 dB per doubling of distance (cylindrical spreading). At short distances near any finite source the geometry is mixed; the inverse square law is most accurate when you are well beyond the physical size of the source.

How do I find the safe stand-off distance from a loud noise source?

Enter the known SPL at a reference distance (e.g. 110 dB at 1 m from an industrial machine) and adjust the target distance until the result falls below your exposure threshold — OSHA sets an 8-hour limit of 90 dB, while NIOSH recommends 85 dB. Note that this gives free-field estimates only; barriers, walls, and personal protective equipment reduce levels further in ways this calculator does not model.

Can I use this calculator in reverse to find the source SPL from a measurement at distance?

Yes — rearrange the formula: SPL₁ = SPL₂ + 20 × log₁₀(d₂ / d₁). Set d₁ to your reference distance (typically 1 m), d₂ to your measurement distance, and SPL₂ to your measured level. The result is the estimated SPL at 1 m, commonly quoted as the sound power level reference for equipment datasheets, assuming free-field conditions hold at your measurement point.

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