Decibel Sound Level

Because the ear handles intensities spanning twelve orders of magnitude, acousticians compress the scale with a logarithm: every 10 dB step means ten times the intensity. The reference I₀ = 10⁻¹² W/m² is the threshold of hearing, defined as 0 dB. A quiet library at 10⁻⁹ W/m² works out to 10 log₁₀(10⁻⁹/10⁻¹²) = 30 dB; a jackhammer at 10⁻² W/m² hits 100 dB — ten million times the library's intensity, yet only 70 dB more on the scale.

The logarithm produces some counterintuitive arithmetic. Two identical 60 dB violins together do not make 120 dB — doubling the intensity adds only 10 log₁₀(2) ≈ 3 dB, for 63 dB total. It takes ten violins to gain a full 10 dB, which the ear perceives as roughly "twice as loud." That is also why workplace noise rules bite so sharply: 85 dB is considered safe for an 8-hour shift, but every 3 dB increase doubles the acoustic energy hitting the ear and halves the safe exposure time.