Question

A wave represented by the equation $y=a\cos (\omega t-kx)$ is superimposed with another wave to form a stationary wave such that the point $x=0$ is a node. The equation of the other wave is

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

Answer 1

The correct option is B $-a\cos (\omega t+kx)$

The stationary wave is formed by the superposition of two identical waves traveling in opposite directions.
The wave, $y=a\cos (\omega t-kx)$ is traveling towards right direction, therefore, the other wave should travel in left direction.
For $y_1=-a\cos (\omega t+kx)$, equation of standing wave becomes,
$y^{\prime }=y+y_1=2a\sin \omega t\sin kx$
As at $x=0$, there is a node
$\Rightarrow y^{\prime }=2a\sin \omega t\sin 0=0$
Hence, correct option is $b$ as only in this option, wave is traveling in left direction.