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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

dB
m
1 m100 m
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 dB4× more intenseMove speaker twice as close
1× (same)0 dBNo changeReference point
2× (double)−6 dB4× less intense6 dB rule of thumb
3×−9.5 dB9× less intenseTriple the distance
10×−20 dB100× less intense1 m → 10 m
100×−40 dB10,000× less intense1 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, intensity (power per unit area) decreases proportionally to 1/r².

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).

Important limitations: The inverse square law applies strictly to a point source in a free field with no reflections. 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.

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.
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