Question

Wave represented by the equation \( y_1 = A \cos(kx - \omega t) \) is superimposed on another wave to form a stationary wave such that the point \( x = 0 \) is a node. The equation representing the wave is given by

• A. \( A \cos(kx + \omega t) \)
• B. \( -A \cos(kx + \omega t) \)
• C. \( A \sin(kx + \omega t) \)
• D. \( -A \sin(kx - \omega t) \)

Hint:

The equation of a stationary wave is formed by the superposition of two waves traveling in opposite directions, and the node condition ensures the opposite phase for the two waves.

Answer 1

The Correct Option is B

When two waves interfere to form a stationary wave with a node at \( x = 0 \), the equation of the resulting wave can be obtained by adding the two individual waves. The general equation for a stationary wave is: \[ y = 2A \cos(kx) \cos(\omega t) \] Given the form of the wave \( y_1 = A \cos(kx - \omega t) \), for \( x = 0 \) to be a node, the second wave must have the opposite sign of the first, resulting in the equation \( -A \cos(kx + \omega t) \). 
Hence, the correct answer is (b).