5.5 The decibel, motivated

The dynamic range of sound intensities we routinely encounter is enormous: thirteen orders of magnitude between the threshold of hearing and the threshold of pain. Linear scales are unusable for this kind of range — a graph from $10^{-12}$ to $10^1$ W/m² is impossible to read.

The fix is to use a logarithmic scale. The decibel (dB) is the universal logarithmic unit of intensity (and amplitude) in acoustics.

Definition

The sound intensity level (SIL) of a wave with intensity $I$, relative to a reference intensity $I_\text{ref}$, is

By convention, $I_\text{ref} = 10^{-12}$ W/m² — roughly the threshold of human hearing at 1 kHz. So:

• $L_I = 0$ dB ↔ $I = 10^{-12}$ W/m² (threshold)
• $L_I = 60$ dB ↔ $I = 10^{-6}$ W/m² (conversational)
• $L_I = 120$ dB ↔ $I = 1$ W/m² (pain)

The 13-order-of-magnitude span becomes a 130 dB range. Manageable.

Pressure vs. intensity dB

For a plane wave, $I \propto P^2$. So $\log_{10}(I/I_\text{ref}) = 2 \log_{10}(P/P_\text{ref})$, and

with $P_\text{ref} = 20\,\mu$Pa — the pressure corresponding to $I_\text{ref}$ in air. The factor of 20 (rather than 10) is because pressure is amplitude-like; we square it to get intensity-like, which then enters log times 10.

Every doubling of pressure is 6 dB. Every doubling of intensity is 3 dB. These two factoids — 6 dB per pressure doubling, 3 dB per intensity doubling — recur constantly.

Useful intuitions

A doubling of distance from a point source drops the intensity by 6 dB (because intensity goes as $1/r^2$ for a point source, and $\log_{10}(1/4) = -0.6$).

A factor of 10 in intensity is 10 dB. A factor of 2 in intensity is 3 dB. A factor of 2 in pressure is 6 dB.

What “dB” alone means

A bare “dB” without a reference is a ratio. “+3 dB” means “intensity doubled.” “-10 dB” means “intensity cut to a tenth.” When you see “100 dB SPL” or “70 dB SIL”, the SPL/SIL marks the reference — usually $P_\text{ref} = 20\,\mu$Pa or $I_\text{ref} = 10^{-12}$ W/m² respectively. Many other reference levels exist in special fields (dBV in electronics, dBA for A-weighted noise levels, dB re. 1 µPa underwater) — always check what the reference is.

Why the ear cares

The minimum audible pressure of $20\,\mu$Pa is seven orders of magnitude smaller than atmospheric pressure. The ear’s amplitude sensitivity threshold is, at peak frequency, roughly $10^{-11}$ m — about the diameter of a single atom. The mechanical system that achieves this is the subject of the Hearing book; what matters here is that the auditory system operates over a dynamic range that demands logarithmic representation, and the decibel is the engineer’s compromise: it is what the ear and brain effectively use anyway, just made explicit.

The decibel is not a physical scale. It is a human-perception scale applied to physical quantities. The next time you see “10 dB louder” — that’s a factor of ten in intensity, but perceptually a doubling of loudness. Both are true; both matter.