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 $x=0.15m$
(2) An antinode occurs at $x=0.3m$
(3) The speed of wave is $15m{s}^{-1}$
(4) The wavelength is $0.2m$

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