Assume that the displacement $(s)$ of air is proportional to the pressure difference $\left ( \Delta p \right )$ created by a sound wave. Displacement (s) further depends on the speed of sound $\left ( \vartheta \right )$ , density of air $\left ( \rho \right )$ and the frequency $\left ( f \right )$ . If $\Delta p\sim 10Pa,\; \vartheta \sim 300 m/s,\; \rho \sim 1\; kg/m^{3}$ and $f-1000\; Hz$ , then s will be of the order pf (take the multiplicative constant to be $1$ )
Option: 1 $\frac{3}{100}mm$
Option: 2 $10\: mm$
Option: 3 $\frac{1}{10}\: mm$
Option: 4 $1\: mm$

Answers (1)

$\begin{array}{l} \because \Delta P_{0}=\mathbf{B} \frac{\omega}{\mathbf{v}} \times \mathbf{S}_{0} \\ \\ \Rightarrow \mathbf{S}=\frac{\Delta \mathbf{P} \times \mathbf{v}}{\mathbf{B} \omega}=\frac{\Delta \mathbf{P} \times \mathbf{v}}{\rho \mathbf{v}^{2} \times \mathbf{2} \pi v} \\ \\ \Rightarrow \mathbf{S} \propto \frac{\Delta \mathbf{P}}{\rho \mathbf{v f}} \\ \\ \Rightarrow \quad \mathbf{S}=\frac{\Delta P}{\rho \mathbf{P}} \\ \\ =\frac{10}{1 \times 300 \times 1000} \\ \\ =\frac{1}{30} \mathrm{~mm} \\ \\ \quad \approx \frac{3}{100} \mathrm{~mm} \end{array}$

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