sound wave

[logistic_guy]

\(\displaystyle \bold{(a)}\) Calculate the maximum displacement of air molecules when a \(\displaystyle 210\)-\(\displaystyle \text{Hz}\) sound wave passes whose intensity is at the threshold of pain \(\displaystyle (120 \ \text{dB})\). \(\displaystyle \bold{(b)}\) What is the pressure amplitude in this wave?

 

[khansaheb]

\(\displaystyle \bold{(a)}\) Calculate the maximum displacement of air molecules when a \(\displaystyle 210\)-\(\displaystyle \text{Hz}\) sound wave passes whose intensity is at the threshold of pain \(\displaystyle (120 \ \text{dB})\). \(\displaystyle \bold{(b)}\) What is the pressure amplitude in this wave?

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[logistic_guy]

\(\displaystyle \bold{(a)}\)

Sound intensity formula is:

\(\displaystyle \beta = 10\log_{10}\frac{I}{I_0}\)

We solve for sound intensity \(\displaystyle I\).

\(\displaystyle 120 = 10\log_{10}\frac{I}{10^{-12}}\)

This gives:

\(\displaystyle I = 1 \ \text{W}\text{/m}^2\)

The maximum displacement is given by:

\(\displaystyle I = 2\pi^2\rho f^2D^2_Mv\)

\(\displaystyle 1 = 2\pi^2(1.29)(210)^2D^2_M(343)\)

This gives:

\(\displaystyle D_M = \textcolor{blue}{5.1 \times 10^{-5} \ \text{m}}\)

 

[logistic_guy]

What is the pressure amplitude in this wave?

Pressure amplitude denoted \(\displaystyle \Delta P_M\) is given by:

\(\displaystyle \Delta P_M = 2\pi \rho v D_M f\)

Plug in numbers.

\(\displaystyle \Delta P_M = 2\pi(1.29)(343)\left(5.1 \times 10^{-5}\right)(210) = 29.78 \ \text{N/}\text{m}^2 = \textcolor{blue}{29.78 \ \text{Pa}}\)