Assume that the displacement(s) of air is proportional to the pressure difference

Question:

Assume that the displacement(s) of air is proportional to the pressure difference $(\Delta \mathrm{p})$ created by a sound wave. Displacement(s) further depends on the speed of sound (v), density of air ( $\rho$ ) and the frequency (f). If $\Delta \mathrm{p} \sim 10 \mathrm{~Pa}, \mathrm{v} \sim 300 \mathrm{~m} / \mathrm{s}, \mathrm{p} \sim 1 \mathrm{~kg} / \mathrm{m}^{3}$ and f $1000 \mathrm{~Hz}$, then $\mathrm{s}$ will be the order of (take multiplicative constant to be 1 )

  1. $10 \mathrm{~mm}$

  2. $\frac{3}{100} \mathrm{~mm}$

  3. $1 \mathrm{~mm}$

  4. $\frac{1}{10} \mathrm{~mm}$


Correct Option: 2

Solution:

$\Delta \mathrm{p}=\mathrm{BkS}_{0}$

$=\rho v^{2} \times \frac{\omega}{v} \times S_{0}$

$\Rightarrow \mathrm{S}_{0}=\frac{\Delta \mathrm{p}}{\rho v \omega}$

$\approx \frac{10}{1 \times 300 \times 1000} \mathrm{~m}$

$=\frac{1}{30} \mathrm{~mm} \approx \frac{3}{100} \mathrm{~mm}$

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