decibels

Meaning: The decibel expresses how intense a sound is relative to a reference, on a logarithmic scale. The ear responds to an enormous range of intensities, so the scale compresses them into small numbers. The reference is the threshold of human hearing, which is 0 dB. Each increase of 10 dB multiplies intensity by 10, and an increase of about 3 dB doubles it.

Equation: β = 10 log(I/I₀), where β is the sound intensity level in decibels (dB, dimensionless), I is the intensity in W/m², and I₀ = 1.0 × 10⁻¹² W/m² is the reference intensity. For a change: β₂ − β₁ = 10 log(I₂/I₁).

On the MCAT: Passages give intensities or levels and ask for the other, usually with powers of ten. A 1000-fold rise in intensity is +30 dB. For a point source in open space, intensity falls as 1/r², so doubling the distance cuts intensity to one-fourth, which is about 6 dB lower. N identical sources together raise the level by 10 log N.

Common confusion: Students treat decibels as a linear scale. 60 dB is not twice 30 dB; it is 1000 times the intensity. Two 50 dB sources together give about 53 dB, not 100 dB. Decibels are a logarithm of an intensity ratio, so convert to intensity before combining sources.