The faintest sounds the human ear can detect at a frequency of 1 000 Hz correspond to an intensity of about $1.00 \times 10^{-12} W / m ^{2}$ —the so-called threshold of hearing. The loudest sounds the ear can tolerate at this frequency correspond to an intensity of about 1.00 W/m²— the threshold of pain. Determine the pressure amplitude and displacement amplitude associated with these two
limits.

Question

The faintest sounds the human ear can detect at a frequency of 1 000 Hz correspond to an intensity of about [latex] 1.00 imes 10^{-12} W / m ^{2} [/latex]—the so-called threshold of hearing. The loudest sounds the ear can tolerate at this frequency correspond to an intensity of about 1.00 W/m²— the threshold of pain. Determine the pressure amplitude and displacement amplitude associated with these two
limits.

Accepted Answer

First, consider the faintest sounds. Using Equation 17.6 and taking v = 343 m/s as the speed of sound waves in air and ρ = 1.20 kg/m³ as the density of air, we obtain

[latex] I=\frac{\Delta P_{\max }^{2}}{2 ho v} [/latex] (17.6).

[latex] \Delta P_{\max }=\sqrt{2 ho v I} [/latex].

[latex] =\sqrt{2\left(1.20 kg / m ^{3} ight)(343 m / s )\left(1.00 imes 10^{-12} W / m ^{2} ight)} [/latex].

[latex] =2.87 imes 10^{-5} N / m ^{2} [/latex].

Because atmospheric pressure is about [latex] 10^{5} N / m ^{2} [/latex], this result tells us that the ear is sensitive to pressure fluctuations as small as 3 parts in [latex] 10^{10} [/latex]!
We can calculate the corresponding displacement amplitude by using Equation 17.4, recalling that )[latex] \omega=2 \pi f [/latex] (see Eqs. 16.3 and 16.9):

[latex] \Delta P_{\max }= ho v \omega s_{\max } [/latex] (17.4).

[latex] f=\frac{1}{T} [/latex] (16.3).

[latex] \omega \equiv \frac{2 \pi}{T} [/latex] (16.9).

[latex] s_{\max }=\frac{\Delta P_{\max }}{ ho v \omega}=\frac{2.87 imes 10^{-5} N / m ^{2}}{\left(1.20 kg / m ^{3} ight)(343 m / s )(2 \pi imes 1000 Hz )} [/latex].

[latex] =1.11 imes 10^{-11} m [/latex].

This is a remarkably small number! If we compare this result [latex] ext { for } s_{\max } ext { with the size of an atom (about } 10^{-1} 10 m ext { ), we see } [/latex] that the ear is an extremely sensitive detector of sound waves.
In a similar manner, one finds that the loudest sounds the human ear can tolerate correspond to a pressure amplitude of 28.7 N/m² and a displacement amplitude equal to [latex] 1.11 imes 10^{-5} m [/latex]

Question 17.2: The faintest sounds the human ear can detect at a frequency ...

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