Decibel (dB)

The decibel (dB) is a logarithmic unit expressing the ratio of two quantities of power. Each 10 dB corresponds to a 10× ratio; 20 dB is 100×; 30 dB is 1000×. The logarithmic compression makes a unit that’s manageable across the enormous dynamic range of physical signals — from a whisper to a jet engine, sounds span 12 orders of magnitude in power, but only 120 dB on the decibel scale.

Formula: dB = 10 × log₁₀(P₁/P₀) for power, or 20 × log₁₀(A₁/A₀) for amplitude (because power scales as amplitude squared). The 10 comes from the “deci” prefix on the original bel unit (named after Alexander Graham Bell, by Bell Labs engineers measuring telephone line attenuation).

Decibels are dimensionless

A bare “dB” value is meaningless without context — it’s always a ratio. To make them concrete, the reference point gets a suffix:

• dBSPL — sound pressure level relative to 20 µPa (the human hearing threshold at 1 kHz).
• dBm — power relative to 1 milliwatt. Standard in RF / wireless engineering.
• dBV — voltage relative to 1 volt. Used in audio gear specs.
• dBu — voltage relative to 0.775 V (the voltage that delivers 1 mW into 600 Ω, historical telephony).
• dBFS — full-scale digital audio. 0 dBFS = the highest value the bit-depth can represent; everything quieter is negative.
• dBA / dBC — sound pressure with frequency weighting to match human hearing sensitivity (A) or flatter low-frequency response (C).

Real-world sound levels

• Threshold of hearing: 0 dBSPL
• Quiet bedroom: 25-30 dBSPL
• Whisper at 1 m: 30 dBSPL
• Office background: 50 dBSPL
• Normal conversation at 1 m: 60 dBSPL
• City traffic at 5 m: 80 dBSPL
• Subway train at platform: 95 dBSPL
• Rock concert front row: 110-120 dBSPL
• Jet engine at 30 m: 140 dBSPL
• Threshold of pain: 130-140 dBSPL

Hearing damage risk starts around 85 dBSPL for sustained exposure (8-hour work shifts). Every additional 3 dB halves the safe exposure time: 88 dB → 4 hours, 91 dB → 2 hours, 94 dB → 1 hour. Modern smartphone apps can estimate ambient dBSPL within ±3 dB using calibrated microphone tables.

Worked example

A speaker rated 90 dB SPL at 1 W / 1 m driven with 100 W produces: 10 × log₁₀(100/1) = 20 dB above the rating, so 110 dB SPL at 1 m. Move to 4 m and inverse-square attenuates 12 dB (6 dB per doubling of distance, twice), giving 98 dB at the listener position. Now compute the safe exposure time: 98 dB is 13 dB above the 85 dB OSHA action level. The exposure halves for every 3 dB increase, so 13 ÷ 3 ≈ 4.3 doublings of risk, which means safe time drops from 8 hours by a factor of 2⁴·³ ≈ 20 — to roughly 24 minutes of continuous exposure before hearing damage risk becomes significant. Concert-goers stand within a few metres of stage stacks pushing 115-120 dB; the 110 dB OSHA “ceiling limit” is breached after a few minutes of an unprotected front-row position, which is why touring sound engineers and seasoned concertgoers wear musician’s earplugs (typically -15 to -25 dB flat attenuation).

When and why it matters

Decibels matter whenever you’re comparing two signals that span orders of magnitude — audio mixing (where a 6 dB difference between two tracks is a doubling of voltage amplitude), RF link budgets (where every interconnect, antenna, and free-space loss contributes a few dB and the sum determines whether the link closes), photography dynamic range (where modern sensors are quoted at 13-14 stops, or 78-84 dB), and any signal-to-noise calculation. The most common misuse is adding dB values directly across uncorrelated channels — 50 dB + 50 dB is not 100 dB, it’s 53 dB (the powers add, not the logs). For two equal sources, doubling the power adds 3 dB; halving subtracts 3 dB; for amplitude, the factor is 6 dB. Memorise “3 dB = ×2 power, 6 dB = ×2 amplitude, 10 dB = ×10 power, 20 dB = ×10 amplitude” and most engineering audio/RF conversations become readable. Reference: OSHA — Occupational Noise Exposure.

Decibels in non-audio domains

In RF engineering, antenna gain and cable loss are quoted in dB. A “3 dB attenuator” halves the signal power. Mobile-phone signal bars correspond loosely to dBm values: ~-50 dBm is full bars; -110 dBm is barely connected. In imaging, dynamic range and signal-to-noise ratios use the same logarithmic scaling. The unifying idea is that perception of intensity (loudness, brightness) is roughly logarithmic to physical magnitude, so a logarithmic unit matches human intuition better than a linear one.