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

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