Question

A travelling wave represented by $y=Asin \left(\right. \omega t - k x \left.\right) \, $ is superimposed on another wave represented by $y=Asin \left(\right.\omega t+kx\left.\right)$ . The resultant is:

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

Answer 1

Answer: (A)

From given progressive waves produced stationary wave
$y_{s w}=A \, \left[\right.sin \left(w t - k x\right) + sin ⁡ \left(\right. w t + k x \left.\right) \left]\right. \, $
$y_{s w}=2Acos k x sin ⁡ w t$
At $x=0,\frac{\lambda }{2}, \, \lambda \ldots \Rightarrow y_{s w}=2a= \, $ Antinode
At $x=\frac{\lambda }{4},\frac{3 \lambda }{4}.. \, \Rightarrow y_{s w}=0=$ Nodes