Question

Due to a point isotropic sonic source, loudness at a point is $L=60\text{dB}$. If density of air is $\rho =\left(15/11\right){\text{kg/m}}^3$ and velocity of sound in air is $v=330\text{m/s}$, the pressure oscillation amplitude at the point of observation is: $[I_0={10}^{-12}{\text{W/m}}^2]$

• A. $0.3{\text{N/m}}^2$
• B. $0.03{\text{N/m}}^2$
• C. $3\times {10}^{-3}{\text{N/m}}^2$
• D. $3\times {10}^{-4}{\text{N/m}}^2$

Answer 1

The correct option is B $0.03{\text{N/m}}^2$

Loudness is given by
$L=60\text{dB}=10{\log }_{10}\frac{I}{I_0}$
where $I_0={10}^{-12}{\text{W/m}}^2$ (threshold of hearing)
$\Rightarrow I=({10}^6\times {10}^{-12}){\text{W/m}}^2={10}^{-6}{\text{W/m}}^2$

Also, we know intensity $I=\frac{{\left(\Delta P_m\right)}^2}{2\rho v}$
where $\Delta P_m$ is the pressure amplitude at that point.
$\rho =15/11{\text{kg/m}}^3,v=330\text{m/s}$ (given)

$\therefore {\left(\Delta P_m\right)}^2=2\rho vI=2\times \frac{15}{11}\times 330\times {10}^{-6}$
$\Rightarrow \Delta P_m=0.03{\text{N/m}}^2$