Loudness spans a trillion-to-one range of power, so the ear is metered on a logarithmic ladder. A point source spreads its energy over a growing sphere — intensity falls as 1/r² — and every rung of the ladder is L = 10·log₁₀(I/I₀). Ten times the power is ten decibels, not ten times the loudness.
source Fletcher, H. & Munson, W. A., “Loudness, Its Definition, Measurement and Calculation,” J. Acoust. Soc. Am. 5, 82 (1933); the decibel = 1/10 bel, named for A. G. Bell (Bell Labs). amber Rendered, not quoted.
A point source of acoustic power P watts. At radius r that power is smeared over a sphere of area 4πr², so the intensity is
I(r) = P / (4πr²) W/m²
L = 10·log₁₀(I / I₀) dB (I₀ = 1e-12 W/m²)
Two levers only: how far energy has spread (1/r²) and how the ear compresses it (log₁₀). The reference I₀ is the threshold of hearing — 0 dB.
The rungs are made of the waves of the-speed-of-sound: what travels at c = √(γRT/M) ≈ 343 m/s is a pressure disturbance, and its intensity is what this ladder meters.
Pressure → intensity → perceived loudness. The decibel is the perceptual scale laid over those waves: −6 dB per distance-doubling, +6 dB per pressure-doubling, +3 dB for a second equal source.
Re-runs the engine's law live and confirms the ladder still reads +10 dB per 10× and −6 dB per doubling. If Red (6) swaps the log for a linear scale, this flips.
Source power P and a listening distance r. Drag them and watch the sphere thin the intensity.
Left: 1/r² falloff (each doubling drops −6 dB). Right: the dB ladder, live marker at the current level.
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“A dB is a fixed loudness — 60 dB is twice 30 dB.” No. The scale is logarithmic and relative. 60 dB is a 1000× greater intensity than 30 dB, and perceived loudness roughly doubles only every +10 dB.
Real failure modes: dB is meaningless without its reference (dB SPL vs dBA vs dBm); free-field 1/r² fails indoors where reflections dominate; coherent sources add in pressure, not power.
“Two equal speakers together are twice as loud: +100%.”
→ Doubling the power is only 10·log₁₀(2) ≈ +3.01 dB — barely noticeable.
“Move twice as far → half the loudness.”
→ Twice as far → a quarter the intensity → −6.02 dB, not −50%.
“Double the pressure → +3 dB.”
→ Intensity goes as pressure squared, so pressure uses the factor 20: 20·log₁₀(2) ≈ +6.02 dB.
Swap the logarithm for a linear scale (L = I/I₀). The trillion-to-one range stops compressing and 10× is no longer +10 dB. The Witness (7) catches it.