Question

A wave disturbance in a medium is described by $\text{y (x,t)}=0.02\text{cos}(50\pi t+\frac{\pi }{2})\text{cos}\left(10\pi x\right),$ where x and y in metre and t is in second.

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

Answer 1

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

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

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