For airborne sound, the decibel scale is defined by: SPL = 20log10[prms/pref], where SPL is sound pressure level in dB (the abbreviation for decibels), prms is the root mean-square pressure amplitude of the sound of interest, and pref = 20 μPa (an international standard). Here, pref is chosen so

Question

For airborne sound, the decibel scale is defined by: $S P L=20 \log _{10}\left[p_{r m s} / p_{r e f}\right]$ , where SPL is sound pressure level in dB (the abbreviation for decibels), $p_{rms}$ is the root mean-square pressure amplitude of the sound of interest, and $p_{ref}$ = 20 μPa (an international standard). Here, $p_{ref}$ is chosen so that 0 dB corresponds to quiet sounds at the nominal threshold of human hearing. Using this information, assess the validity of (15.11) for sounds with SPL = 30 dB (soft whispering), 60 dB (normal conversation), 90 dB (noisy factory), 115 dB (good seat at a rock concert), 130 dB (near an aircraft engine), and 160 dB (inside of an automobile exhaust system) at atmospheric pressure.

$p^{\prime} / \rho_0 c^2 \ll 1$ . (15.11)

[latex]p^{\prime} / \rho_0 c^2 \ll 1[/latex]. (15.11)

Accepted Answer

The definition of SPL can be inverted to find [latex]p_{rms}=p_{ref}10^{(SPL/20)}, ext{ and } ho_0c^2\equiv\gamma p_0\approx140 \mathrm{kPa}[/latex] at atmospheric pressure. Thus, for SPL = 30 dB, (15.11) becomes:

[latex]p_{rms}/ ho_0c^2=(20 \mu\mathrm{Pa})10^{(30/20)}/(140 \mathrm{kPa})=4.5 imes10^{-9}\ll1[/latex],

and the small amplitude requirement for linear acoustics is well satisfied. The remaining SPL values lead to the following numbers:

[latex]60 \mathrm{dB}:\quad p_{rms}/ ho_0c^2=1.4 imes10^{-7}[/latex],

[latex]90 \mathrm{dB}:\quad p_{rms}/ ho_0c^2=4.5 imes10^{-6}[/latex],

[latex]115 \mathrm{dB}:\quad p_{rms}/ ho_0c^2=8.0 imes10^{-5}[/latex],

[latex]130 \mathrm{dB}:\quad p_{rms}/ ho_0c^2=4.5 imes10^{-4}[/latex],

[latex]160 \mathrm{dB}:\quad p_{rms}/ ho_0c^2=1.4 imes10^{-2}[/latex].

Thus, even at decibel levels that cause hearing damage (130 dB) or deafness (160 dB), the smallamplitude requirement for linear acoustics is met. Therefore, the linear theory of acoustics applies throughout (and somewhat beyond) the amplitude range of human hearing.

Question 15.2: For airborne sound, the decibel scale is defined by: SPL = 2...

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