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 \( x,y \) are in meters and \( t \) in seconds. Which statements are correct?

• A. A node occurs at \( x=0.15\,\text{m} \)
• B. An antinode occurs at \( x=0.3\,\text{m} \)
• C. The speed of the wave is \( 4\,\text{m/s} \)
• D. The wavelength of the wave is \( 0.2\,\text{m} \)

Hint:

The wave is a standing wave of the form \( \cos\omega t \cos kx \). Nodes are where \( \cos kx = 0 \) and antinodes where \( \cos kx = \pm 1 \). Speed is \( \omega / k \).

Answer 1

The Correct Option is A, B, D

Step 1: Plan:
We find the wavelength and frequency first. Then we place nodes and antinodes using the wavelength, and get the speed from frequency times wavelength.

Step 2: Frequency and wavelength:
The time part has angular frequency \(50\pi\), so \(f = 50\pi / 2\pi = 25\) Hz. The position part \( \cos(10\pi x) \) repeats when \(10\pi x\) changes by \(2\pi\), which is a change of 0.2 m. So \(\lambda = 0.2\) m and (D) is true.

Step 3: Speed:
Speed is frequency times wavelength.
\[v = f\lambda = 25 \times 0.2 = 5\ \text{m/s}\]
This is not 4 m/s, so (C) is false.

Step 4: Position of nodes and antinodes:
At \( x = 0 \) the position factor is 1, so there is an antinode at the origin. Antinodes then come every half wavelength, which is 0.1 m. So antinodes sit at 0, 0.1, 0.2, 0.3 m and so on. A node lies exactly midway between two antinodes, a quarter wavelength (0.05 m) from the origin, then every 0.1 m: 0.05, 0.15, 0.25 m.

Step 5: Match the statements:
0.15 m is in the list of nodes, so (A) is true. 0.3 m is in the list of antinodes, so (B) is true. \[ \boxed{\text{(A), (B) and (D)}} \]