What is Sound level? — Physics

Definition 21.9 High School Physics · Chapter 21 — Sound and Acoustics

The ear spans twelve powers of ten, so intensities are quoted logarithmically. The sound level of $I$ is

L = 10 \log_{10}(I/I_0), \qquad I_0 = 1.0 \times 10^{-12}\,\mathrm{W}/\mathrm{m}^{2},

in decibels ( $\mathrm{dB}$ ): hearing threshold $0\,\mathrm{dB}$ , pain near $120\,\mathrm{dB}$ .

A ladder of everyday levels: each 20\, dB step is a hundredfold intensity jump; beyond 85\, dB (red zone) exposure damages hearing. — A ladder of everyday levels: each $20\,\mathrm{dB}$ step is a hundredfold intensity jump; beyond $85\,\mathrm{dB}$ (red zone) exposure damages hearing.

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Definition 7.5 University Physics — Year 2 · Chapter 7 — Sound Waves in Fluids

The sound level of a wave of mean intensity $I$ is

L = 10\log_{10}\frac I{I_0} \ \text{dB} , \qquad I_0 = 1 \times 10^{-12}\,\mathrm{W}/\mathrm{m}^{2}

— the threshold of hearing at $1\,\mathrm{kHz}$ , corresponding in air to the pressure amplitude $p_0 \approx 2 \times 10^{-5}\,\mathrm{Pa}$ (rms) $= 2 \times 10^{-10}$ atmospheres. Doubling the intensity adds $3\,\mathrm{dB}$ ; ten times, $10\,\mathrm{dB}$ ; a hundred times the pressure amplitude, $40\,\mathrm{dB}$ . Quiet room $30\,\mathrm{dB}$ , conversation $60\,\mathrm{dB}$ , busy street $80\,\mathrm{dB}$ , rock concert $110\,\mathrm{dB}$ , pain $120\,\mathrm{dB}$ ( $I = 1\,\mathrm{W}/\mathrm{m}^{2}$ , $p_m = 29\,\mathrm{Pa}$ ).

Left: in a plane progressive sound wave the overpressure and the fluid velocity are in phase, and the displacement lags by a quarter period. Right: the decibel scale — each 20\, dB step multiplies the intensity by a hundred and the pressure amplitude by ten. — Left: in a plane progressive sound wave the overpressure and the fluid velocity are in phase, and the displacement lags by a quarter period. Right: the decibel scale — each $20\,\mathrm{dB}$ step multiplies the intensity by a hundred and the pressure amplitude by ten.

Examples

Example 7.6 (How small a sound is)

At the threshold, $1\,\mathrm{kHz}$ : $p_m = 2.8 \times 10^{-5}\,\mathrm{Pa}$ , $v_m = p_m/Z = 7 \times 10^{-8}\,\mathrm{m}/\mathrm{s}$ , and the displacement amplitude $\xi_m = v_m/\omega = 1 \times 10^{-11}\,\mathrm{m}$ — a tenth of an atomic diameter: the eardrum detects motions smaller than an atom. At $120\,\mathrm{dB}$ , $\xi_m = 11\,\text{µ}\mathrm{m}$ , still invisible, and $p_m/P_0 = 3 \times 10^{-4}$ : even the loudest sounds are tiny perturbations, which is why the linear theory works so well.

Example 7.8 (A siren, a conversation)

A $10\,\mathrm{W}$ siren: $I = 10/4\pi r^2$ , $0.8\,\mathrm{W}/\mathrm{m}^{2}$ at $1\,\mathrm{m}$ ( $119\,\mathrm{dB}$ , pain), $79\,\mathrm{dB}$ at $100\,\mathrm{m}$ , $59\,\mathrm{dB}$ at $1\,\mathrm{km}$ — still audible over the city. A voice radiates about $10\,\text{µ}\mathrm{W}$ : $60\,\mathrm{dB}$ at $1\,\mathrm{m}$ , and $40\,\mathrm{dB}$ at $10\,\mathrm{m}$ . Air absorbs sound too (more at high frequency, a few decibels per hundred metres at $10\,\mathrm{kHz}$ ), which is why distant thunder rumbles low.

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Examples

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