Question

A travelling wave represented by y(x, t) = a sin (kx – $\omega$t) is superimposed on another wave represented by y(x, t) = a sin (kx + $\omega$t). The resultant is a

• A. Standing wave having nodes at x =$\left( n + \frac{1}{2} \right)\frac{\lambda}{2};$ n = 0, 1, 2, ....
• B. Standing wave having nodes at x = $\frac{n\lambda}{2}$; n = 0, 1, 2, ...
• C. Wave travelling along + x direction
• D. Wave travelling along – x direction

Answer 1

Answer: (B)

Standing wave having nodes at x = $\frac{n\lambda}{2}$; n = 0, 1, 2, ...

Answer 2

According to the principle of superposition, the resultant wave is

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

$2a\sin kx\cos\omega t$ …. (i)

It represent a standing wave. In the standing wave, there will be nodes (where amplitude is zero) and antinodes (where amplitude is largest).

For Eq. (i), the positions of nodes are given by

Sin k x = 0

$\Rightarrow kx = n\pi;$ n = 0,1,2, ……..

Or $\frac{2\pi}{\lambda}x = n\pi,$ n=0,1,2,……..

Or $x = \frac{n\lambda}{2};$ n= 0,1,2, ……….

In the same way,

From Eq. (i), the positions of antinodes are given by

$\left| \sin kx \right| = 1$

$\Rightarrow kx = \left( n + \frac{1}{2} \right)\pi;n = 0,1,2,.......$

Or $\frac{2\pi x}{\lambda} = \left( n + \frac{1}{2} \right)\pi;n = 0,1,2,.......$

Or $x = \left( n + \frac{1}{2} \right)\frac{\lambda}{2};n = 0,1,2,......$