Q153 / 155 · WBJEE 2025 · Physics

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 $x, y$ are in meters and $t$ is in second. Which statement(s) is/are correct?

  • Correct answer
    A
    A node occurs at $x=0.15 \mathrm{~m}$
  • Correct answer
    B
    An antinode occurs at $x=0.3 \mathrm{~m}$
  • C
    The speed of the wave is $4 \mathrm{~m} / \mathrm{sec}$
  • Correct answer
    D
    The wavelength of the wave is 0.2 m

Solution

$\cos 10 \pi x=0 \rightarrow$ Node $\begin{aligned} & 10 \pi \mathrm{x}=\mathrm{n} \pi+\frac{\pi}{2} \\ & 10 \pi \mathrm{x}=\mathrm{n} \pi+\frac{1}{2} \\ & \mathrm{x}=0, \frac{1}{20} \mathrm{~m}, \frac{3}{20} \mathrm{~m} \\ & \cos \pi \mathrm{x}= \pm 1 \rightarrow \text { Antinode } \\ & 10 \pi \mathrm{x}=\mathrm{n} \pi, \mathrm{x}=0, \frac{1}{10} \mathrm{~m}, \frac{2}{10} \mathrm{~m}, \frac{3}{10} \mathrm{~m} \\ & \text { wave speed }(\mathrm{v})=\frac{\omega}{\mathrm{k}}=\frac{50 \pi}{10 \pi}=5 \mathrm{~m} / \mathrm{s}, \mathrm{k}=10 \pi, \frac{2 \pi}{\lambda}=10 \pi, \lambda=\frac{2}{10}=0.2 \mathrm{~m} \end{aligned}$

Hence, the correct answer is options: 'A', 'B', 'D'.

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