Question

A sound wave in air has a frequency of $30Hz$ and a displacement amplitude of $6.0\times {10}^{-3}mm$. For this sound wave calculate the sound intensity level (in dB)
Speed of sound = $344m/s$ and density of air = $1.2kg/m^3$

Answer 1

Intensity $I$ of a plane wave of amplitude $A$ , frequency $f$ , travelling with a velocity of $v$ in a medium of density $\rho$ is given by ,

$I=2{\pi }^2{f}^2{A}^2\rho v$ ,

we have , $v=344m/s,\rho =1.2kg/m^3,f=30Hz,A=6.0\times {10}^{-3}mm=6.0\times {10}^{-6}m$ ,

therefore , $I=2\times {3.14}^2\times {30}^2\times (6.0\times {10}^{-6}{)}^2\times 1.2\times 344=2.63\times {10}^{-4}W/m^2$ ,

now sound intensity level (loudness ) is given by ,

$L=lo{g}_{10}I/I_0$ ,

where $I_0={10}^{-12}W/m^2$ , (zero level of intensity)

therefore , $L=lo{g}_{10}2.63\times {10}^{-4}/{10}^{-12}=lo{g}_{10}2.63\times {10}^8$ ,

or $L=lo{g}_{10}2.63+lo{g}_{10}{10}^8=8.419bel=84.19dB$