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.

• A. A node occurs at $x = 0.15\, m$
• B. An antinode occurs at $x = 0.3\, m$
• C. The speed of wave is $5 ms^{-1} $
• D. The wavelength of wave is $0.2\, m$

Answer 1

Answer: (A), (B), (C), (D)

It is given that y(x, t)=0.02\cos $(50 \pi t+ \pi /2)\cos (10 \pi x) $
$ \cong A \, \cos(\omega t+ \pi/2)\cos \, kx $
Node occurs when $kx=\frac {\pi }{2}, \frac {3 \pi }{2} \, etc. $
$10 \pi x= \frac {\pi}{2}, \frac {3 \pi }{2} \Rightarrow x=0.05\,m,0.15\,m$ option (a)
Antinode occurs when $kx= \pi ,2 \pi , 3 \pi \, etc. $
$10 \pi x= \pi ,2 \pi , 3 \pi \, etc. $
$\Rightarrow x=0.1\,m,0.2\,m,0.3\,m$ option (b)
Speed of the wave is given by,
$v=\frac {\omega}{k}= \frac {50 \pi }{10 \pi }=5 \, m/s$ option (c)
Wavelength is given by,
$\lambda=\frac {2 \pi}{k}= \frac {2 \pi}{10 \pi }= \left(\frac {1}{5}\right)m=0.2 m$ option(d)