Question

A source of sound of frequency $50Hz$ is producing longitudinal waves in air. The amplitude of vibrations of air particles is 5mm and the speed of the wave is $330m{s}^{-1}$. Write the equation of the wave.

• A. $y=5\times {10}^{-3}sin[100\pi t-\frac{100\pi x}{33}]$
• B. $y=0.05sin[220\pi t-\frac{\pi x}{3}]$
• C. $y=0.05sin[110\pi t-\frac{2\pi x}{3}]$
• D. $y=0.05sin[110\pi t-\frac{\pi x}{3}]$

Answer 1

Answer: (A)

$y=5\times {10}^{-3}sin[10\pi t-\frac{100\pi x}{33}]$

Given,
$f=50Hz$
$a=5mm=5\times {10}^{-3}m$
$v=330m{s}^{-1}$
Now $\omega =2\pi f=2\pi \times 50=100\pi rad{s}^{-1}$
Wavelength $=\lambda =\frac{v}{f}=\frac{330}{50}=6.6m$
Angular wavenumber $=k=\frac{2\pi }{\lambda }=\frac{2\pi }{6.6}=\frac{10\pi }{33}rad{m}^{-1}$
The equation of the wave can be written as

$y=asin(\omega t-kx)$
$y=5\times {10}^{-3}sin[100\pi t-\frac{10\pi x}{33}]$
x and y are in metre t in seconds.
Ans: $y=5\times {10}^{-3}sin[100\pi t-\frac{10\pi x}{33}]$.