Question

The equation of a stationary wave is \( y = 20 \sin(\pi x) \cos(\omega t) \), where \( x, y \) are in metre and \( t \) is in second. The distance between a node and its adjacent antinode is:

• A. \( 25 \, \text{cm} \)
• B. \( 100 \, \text{cm} \)
• C. \( 50 \, \text{cm} \)
• D. \( 200 \, \text{cm} \)

Hint:

Distance between consecutive nodes or antinodes is \( \frac{\lambda}{2} \). Distance between a node and an adjacent antinode is \( \frac{\lambda}{4} \).

Answer 1

The Correct Option is C

Step 1: Determine the wave number and wavelength.
Comparing \( \sin(\pi x) \) with \( \sin(\frac{2\pi}{\lambda} x) \), we get \( \frac{2\pi}{\lambda} = \pi \), so \( \lambda = 2 \, \text{m} \).
Step 2: Identify the distance between a node and an adjacent antinode.
The distance between a node and an adjacent antinode in a stationary wave is \( \frac{\lambda}{4} \).
Step 3: Calculate the distance.
Distance \( = \frac{2 \, \text{m}}{4} = 0.5 \, \text{m} \)
Step 4: Convert to centimeters. Distance \( = 0.5 \, \text{m} \times 100 \, \text{cm/m} = 50 \, \text{cm} \)