Question

A sound wave of wavelength $40\text{cm}$ travels in air. If the difference between maximum and minimum pressures at a given point is $1.0\times {10}^{-3}{\text{N/m}}^2$, then find the amplitude of vibrations of the particles of the medium.
[Bulk modulus of air is $1.4\times {10}^5{\text{N/m}}^2$]

• A. $3.3\times {10}^{-10}\text{m}$
• B. $2.3\times {10}^{-10}\text{m}$
• C. $4.3\times {10}^{-10}\text{m}$
• D. $5.3\times {10}^{-10}\text{m}$

Answer 1

The correct option is B $2.3\times {10}^{-10}\text{m}$

Let $P$ be the atmospheric pressure.
Difference between maximum and minimum pressures at a given point,
$\left[\right(P+P_o)-(P-P_o\left)\right]=1.0\times {10}^{-3}{\text{N/m}}^2$
$\Rightarrow 2P_o=1.0\times {10}^{-3}{\text{N/m}}^2$
$\Rightarrow P_o=0.5\times {10}^{-3}{\text{N/m}}^2$
We know the relation between maximum value of pressure in the medium and displacement of the particles in the medium is given by,
$P_o=B{A}_{o}k$
$\Rightarrow A_o=\frac{P_o}{Bk}=\frac{\lambda P_o}{2\pi B}$
$\Rightarrow A_o=\frac{40\times {10}^{-2}\times 0.5\times {10}^{-3}}{2\times 3.14\times 1.4\times {10}^5}$
$\Rightarrow A_o=2.3\times {10}^{-10}\text{m}$