Question

Find the amplitude of motion of the air in the path of a $40\text{dB},1000\text{Hz}$ sound wave.
[Assume that the density of air is $1.3{\text{kg/m}}^3$ and speed of sound in $340\text{m/s}$].

• A. $1\text{cm}$
• B. $1\mu \text{m}$
• C. $1\text{nm}$
• D. $1\text{mm}$

Answer 1

The correct option is C $1\text{nm}$

Given that,
$\rho =1.3{\text{kg/m}}^3,v=340\text{m/s},f=1000\text{Hz}\&\beta =40\text{dB}$.

From the relation between loudness and the intensity of sound wave,
$\beta =10{\log }_{10}\left(\frac{I}{I_0}\right)$
From the data given,
$40=10{\log }_{10}\left(\frac{I}{{10}^{-12}}\right)$
$\Rightarrow I={10}^{-12}\times {10}^4={10}^{-8}{\text{W/m}}^2$

Using Intensity formula,
$I=\frac{1}{2}\rho {\omega }^{2}v{A}^2$
$A=\sqrt{\frac{2I}{\rho {\omega }^{2}v}}=\sqrt{\frac{2I}{\rho (2\pi f{)}^{2}v}}$

Substituting the data, we get

$A=\sqrt{\frac{2\times {10}^{-8}}{1.3\times (2\pi \times 1000{)}^2\times 340}}\approx {10}^{-9}\text{m}\text{or}1\text{nm}$