Question

A wave represented by the equation $y=A\sin (kx-\omega t)$ 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. $-A\sin (kx+\omega t)$
• B. $A\cos (kx-\omega t)$
• C. $A\sin (kx+\omega t)$
• D. $A\cos (kx+\omega t)$

Answer 1

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

The given wave has the wave function,
$y=A\cos (kx+\omega t)$
Let, the other wave has the wave function $y^{\prime }$.
At node i.e. stationary point, $x=0$ and the sum of individual wave functions, $y$ and $y^{\prime }$ must be zero at $x=0$ at all times.
Therefore,
$y(0,t)=y+y^{\prime }=0$
$\Rightarrow y^{\prime }=-y$
Therefore,
$y^{\prime }=-A\cos (kx+\omega t)$
The first (-) shows that the second wave is like an inverted wave with respect to the first wave. This is the required equation of the other wave.