The Decibel: Why Loudness Lives on a Logarithmic Scale

From bel to decibel: a ratio, not a unit

The scale began in the 1920s at Bell Telephone Laboratories, where engineers needed to describe power losses along telephone lines that stretched over hundreds of kilometres. Signal power fell off geometrically, so a linear measure was useless; a logarithmic one turned multiplicative loss into additive bookkeeping. They defined the bel — honouring Alexander Graham Bell — as the base-10 logarithm of a power ratio:

• 1 bel = log₁₀(P₁/P₀)
• The bel proved too coarse for everyday use, so the working unit became one tenth of it: the decibel (dB), with 1 bel = 10 dB.

Hence the canonical definition for any power-like quantity:

• L = 10·log₁₀(P/P₀) dB, where P₀ is a chosen reference power.

Because it is a logarithm of a ratio, a decibel is dimensionless — it carries no watts, no pascals, no volts. A bare "60 dB" is physically meaningless until you name the reference. In acoustics the reference is standardised so that 0 dB corresponds to the human threshold of hearing at 1 kHz. The reason the log is multiplied by 10 (and by 20 for pressure) is exactly what lets additive dB values represent multiplicative changes in energy — cascade three lossy amplifiers of −3 dB, −6 dB, −2 dB and the total is simply −11 dB.

The governing equation and its two faces

For sound, the physical quantity that actually carries energy is the intensity I — the time-averaged acoustic power crossing unit area, in W/m². The sound intensity level is

• L_I = 10·log₁₀(I/I₀) dB, with I₀ = 10⁻¹² W/m².

But microphones measure pressure, not power. For a travelling plane wave the acoustic intensity relates to the root-mean-square pressure p by I = p²/(ρc), where ρc is the characteristic acoustic impedance of air (ρ ≈ 1.21 kg/m³, c ≈ 343 m/s, so ρc ≈ 413 Pa·s/m). Because I ∝ p², the logarithm picks up a factor of two:

• L_p = 10·log₁₀(p²/p₀²) = 20·log₁₀(p/p₀) dB, with p₀ = 20 μPa.

The factor 20 for pressure versus 10 for intensity is not two different scales — it is one scale seen through the square-law link between a field amplitude and the power it carries. The references were deliberately matched: plugging p₀ = 20 μPa into I = p²/(ρc) gives I ≈ (20×10⁻⁶)²/413 ≈ 9.7×10⁻¹³ W/m² ≈ 10⁻¹² W/m². So 0 dB SPL and 0 dB intensity level coincide at the threshold of hearing, by design.

Why the ear demands a logarithm

The single most important justification for the log scale is the enormous dynamic range of hearing. From the faintest audible rustle (≈10⁻¹² W/m²) to the onset of pain (≈1–10 W/m²) spans twelve to thirteen orders of magnitude in intensity. No linear meter can display that: if the pain threshold sat at the far end of a one-metre ruler, the threshold of hearing would be 10⁻¹² m from the origin — about a hundredth of the width of a single atom.

Remarkably, perception itself is roughly logarithmic. The Weber–Fechner law (Ernst Weber, 1830s; Gustav Fechner, 1860) states that the smallest perceptible change in a stimulus is proportional to the stimulus already present — Δsensation ∝ ΔI/I. Integrating gives a perceived loudness that grows with log I, precisely the form of the decibel. So a scale built for telephone engineers turned out to mirror psychophysics:

• The just-noticeable difference in loudness is about 1 dB under good conditions.
• A +10 dB change (10× the intensity) is perceived as roughly "twice as loud" — the basis of the psychoacoustic sone scale.
• Doubling the number of identical, incoherent sources (two violins instead of one) adds only +3 dB, which is barely a noticeable change — a fact that surprises every new orchestra member.

The 3 dB and 6 dB rules, worked out

Two arithmetic facts make decibels quick to reason about. First, doubling the power adds 3 dB:

• 10·log₁₀(2) = 10 × 0.301 = 3.01 dB.
• Halving power gives −3.01 dB — the famous "half-power point" that defines the −3 dB bandwidth of a filter or the corner frequency of an RC circuit.

Second, doubling the pressure (or voltage, or any field amplitude) adds 6 dB, because power scales as amplitude squared:

• 20·log₁₀(2) = 20 × 0.301 = 6.02 dB.

These combine cleanly with the geometry of radiation. A point source in free space spreads its power over an expanding sphere of area 4πr², so intensity falls as the inverse-square law, I ∝ 1/r². Doubling the distance quarters the intensity:

• 10·log₁₀(1/4) = −6.02 dB per doubling of distance.

That is why moving from 1 m to 8 m from a loudspeaker (three doublings) drops the level by about 18 dB. A quick worked example: a machine reads 85 dB at 1 m; at 4 m (two doublings) it reads 85 − 12 = 73 dB — comfortably below the 8-hour occupational exposure limit.

Real numbers: the acoustic landscape

Anchoring the scale to familiar sounds shows how much physical range each 10 dB step buys. Every +10 dB is a factor of 10 in intensity; every +20 dB a factor of 100:

• 0 dB — threshold of hearing, I = 10⁻¹² W/m², p = 20 μPa (finer than thermal air-molecule jostling in a quiet room).
• 30 dB — a whisper or quiet library; I = 10⁻⁹ W/m².
• 60 dB — normal conversation; I = 10⁻⁶ W/m², a million times the threshold intensity yet only a millionth of a watt per square metre.
• 90 dB — heavy traffic, lawnmower; sustained exposure risks hearing damage. I = 10⁻³ W/m².
• 120 dB — jet takeoff nearby, threshold of pain; I = 1 W/m², p = 20 Pa.
• 194 dB — the theoretical maximum for an undistorted sound wave in Earth's atmosphere. Here the RMS pressure ≈ 10⁵ Pa equals atmospheric pressure, so the rarefaction phase reaches a perfect vacuum; louder "sounds" become shock waves, not sinusoids.

The 1883 Krakatoa eruption is estimated to have produced ≈172 dB at 160 km — its pressure pulse circled the globe several times and was recorded on barographs worldwide, a vivid reminder that decibels above ~194 describe nonlinear blast physics.

A-weighting, adding decibels, and other subtleties

The ear is not equally sensitive at all frequencies — it is most acute near 3–4 kHz and quite deaf below 100 Hz. To make a single number correlate with perceived loudness, meters apply A-weighting, a frequency filter modelled on the equal-loudness contours (Fletcher–Munson, 1933). Results are quoted in dBA; occupational limits such as the common 85 dBA / 8-hour rule are A-weighted.

Decibels also do not add arithmetically, a persistent misconception. Two independent 60 dB sources do not make 120 dB — you must sum intensities, not levels:

• Convert each level to intensity: 60 dB → I = 10⁶ · I₀ each.
• Add: I_total = 2 × 10⁶ · I₀.
• Convert back: L = 10·log₁₀(2×10⁶) = 63.01 dB, i.e. just +3 dB.

Two further subtleties trip people up. Coherent versus incoherent sources: identical loudspeakers driven in phase can add in pressure (+6 dB) rather than in power (+3 dB) where their waves constructively interfere. And a negative decibel value is perfectly legitimate — it simply means the quantity is below the reference; anechoic-chamber noise floors are routinely quoted at −5 to −10 dBA. The decibel is a ruler with a chosen zero, not a floor at silence.

Power-based vs field-based decibels: the factor of 10 versus 20, and why they agree.

Quantity	Formula	Reference	+10 dB means	Doubling gives
Sound intensity (power/area)	L = 10·log₁₀(I/I₀)	I₀ = 10⁻¹² W/m²	10× the intensity	+3.01 dB
Sound pressure (field)	L = 20·log₁₀(p/p₀)	p₀ = 20 μPa	10× the pressure	+6.02 dB
Electrical power	L = 10·log₁₀(P/P_ref)	1 mW → dBm	10× the power	+3.01 dB
Voltage/amplitude	L = 20·log₁₀(V/V_ref)	context-set	10× the voltage	+6.02 dB