Decibels

The decibel is the most common logarithmic scale in practice: it measures the ratio between a quantity and a reference level.

Definition — Decibel

Given a reference level $P_0$, the level in decibels of a quantity $P$ (power, intensity, etc.) is $L_P = 10\,\log_{10}\!\left(\frac{P}{P_0}\right) \quad\text{dB.}$ For amplitudes (pressure, voltage), which are squared to give the power, one uses $L_A = 20\,\log_{10}(A/A_0)$ dB.

Example — Sound intensity

In acoustics $P_0 = 10^{-12}$ W/m$^2$ is the threshold of hearing. A normal conversation has intensity $P\approx 10^{-6}$ W/m$^2$: $L = 10\log_{10}\!\left(\frac{10^{-6}}{10^{-12}}\right) = 10\log_{10}(10^6) = 10\cdot 6 = 60\ \text{dB}.$ Typical levels: whisper $30$ dB; office $50$ dB; vacuum cleaner $70$ dB; rock concert $110$ dB; threshold of pain $130$ dB.

Property — Additivity of decibels

Doubling the intensity adds $10\log_{10} 2\approx 3$ dB. Multiplying it tenfold adds $10$ dB. A hundredfold, $+20$ dB.

When combining independent sources one adds the intensities (linearly), not the decibels. For two equal sources of $60$ dB each, the total intensity is $2P$, hence $L = 60 + 3 = 63$ dB (not $120$ dB).

Links

Topics: Logarithmic function
Concepts: Decibel · Logarithm · Logarithmic scale
Functions: Logarithmic function
Methods: Decibel
Skills: Calculate · Model