Question

A wave disturbance in a medium is described by $y(x, t)=0.02 \cos \left(50 \pi t+\frac{\pi}{2}\right) \cos (10 \pi x)$, where $x$ and $y$ are in metre and $t$ is in second ?
(1) A node occurs at $x=0.15 \,m$
(2) An antinode occurs at $x=0.3\, m$
(3) The speed of wave is $15\, ms ^{-1}$
(4) The wavelength is $0.2 \,m$

• A. 1 and 2 are correct
• B. 2 and 3 are correct
• C. 1 and 4 are correct
• D. 1,2 and 4 are correct

Answer 1

Answer: (D)

In the given case,
$y(x, t)=0.02 \cos \left(50 \pi t+\frac{\pi}{2}\right) \cos (10 \pi x)$
$\cong A \cos \left(\omega t+\frac{\pi}{2}\right) \cos k x$
Node occurs when
$k x=\frac{\pi}{2}, \frac{3 \pi}{2},$ etc.,
$10 \pi x=\frac{\pi}{2}, \frac{3 \pi}{2} $
$\Rightarrow x=0.05\, m , 0.15 \,m$
Antinode occurs when
$k x=\pi, 3 \pi$ etc.,
$\Rightarrow 10 \pi x=\pi, 3 \pi$
$ x=0.1\, m, 0.3\, m$
Speed of wave is given by
$v=\frac{\omega}{k}=\frac{50 \pi}{10 \pi}=5\, ms ^{-1}$
Wavelength is given by
$\lambda=\frac{2 \pi}{k}=\frac{2 \pi}{10 \pi} $
$=\frac{1}{5} m=0.2\, m$