Question

A sound wave of wavelength $50\text{cm}$ travels in a gas medium. If the difference between the maximum and minimum pressures at a given point is $5\times {10}^{-3}{\text{N/m}}^2$ and the amplitude of vibration of the particles of the medium is $5\times {10}^{-10}\text{m}$, find bulk modulus of the medium.

Answer 1

Given,

Wavelength, $\lambda =50\text{cm}$

Displacement amplitude, $A_0=5\times {10}^{-10}\text{m}$

We know that the relation between maximum pressure in the medium and maximum displacement of the particles in the medium is given by,

$P_o=B{A}_{o}k$

where, $P_o$ is the maximum pressure, $B$ is the bulk modulus, $A_o$ is the maximum displacement and $k$ is the wave number.

$P_o=\frac{P_{max}-P_{min}}{2}$

$P_o=\frac{5\times {10}^{-3}}{2}$

$P_o=2.5\times {10}^{-3}{\text{N/m}}^2$

Also, $k=\frac{2\pi }{\lambda }=\frac{2\pi }{0.5}=4\pi$

Hence, $B=\frac{P_0}{k{A}_0}$

$B=\frac{2.5\times {10}^{-3}}{4\times 3.14\times 5\times {10}^{-10}}$

$B=3.98\times {10}^5{\text{N/m}}^2$

$B\approx 4\times {10}^5{\text{N/m}}^2$