Question

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

• A. $\frac{3}{100}\text{mm}$
• B. $10\text{mm}$
• C. $\frac{1}{10}\text{mm}$
• D. $1\text{mm}$

Answer 1

The correct option is A $\frac{3}{100}\text{mm}$

As we know,
Pressure amplitude,
$\Delta P_0=s_{0}KB=S_0\times \frac{\omega }{V}\times \rho V^2$
$[\because K=\frac{\omega }{V},V=\sqrt{\frac{B}{\rho }}]$
$\Rightarrow s_0=\frac{1}{2\pi }\frac{\Delta P_0}{\rho Vf}$

Taking propotionality constant as 1,
Order of $s_0\approx \frac{10}{1\times 300\times 1000}\text{m}=\frac{1}{30}\text{mm}\approx \frac{3}{100}\text{mm}$