Question

A wave $y=a\sin (\omega t-kx)$ on a string meets with another wave producing a node at $x=0$. Then the equation of the unknown wave is:

• (A) $y=a\sin (\omega t+kx)$
• (B) $y=-a\sin (\omega t+kx)$
• (C) $y=a\sin (\omega t-kx)$
• (D) $y=-a\sin (\omega t-kx)$

Answer 1

The Correct Option is B

To find the equation of the unknown wave, let's first understand the given situation clearly. We have a wave on a string represented by:

y = a \sin(\omega t - kx)

This wave meets another wave to produce a node at x = 0. A node indicates that the two waves are interfering destructively, meaning their displacements cancel each other out at that point.

In wave interference, for two waves to form a node, their displacements must be equal in magnitude but opposite in phase at the node point. The concept of superposition tells us that when two waves overlap, their amplitudes add together. Therefore, at the position of the node, the total displacement is zero.

Let's denote the unknown second wave as y_2. For destructive interference to occur at x = 0, we must have:

a \sin(\omega t - k \times 0) + y_2 = 0 \quad \Rightarrow \quad a \sin(\omega t) + y_2 = 0

This implies that:

y_2 = -a \sin(\omega t)

If we propose that the second wave is traveling in the positive x-direction, its equation would be of the form y_2 = \pm a \sin(\omega t + kx).

Choosing the proper sign to ensure destructive interference at x = 0, it must be:

y_2 = -a \sin(\omega t + kx)

This matches with option (B):

y = -a \sin(\omega t + kx)

Therefore, the equation of the unknown wave is correctly given by option (B).