Question

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

• A. A standing wave having nodes at $x=\left(n+\frac{1}{2}\right)\frac{\lambda }{2}$ , $n=0,1,2,\dots$
• 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
$y_{sw}=A[sin\left(wt-kx\right)+sin(wt+kx)]$
$y_{sw}=2Acoskxsinwt$
At $x=0,\frac{\lambda }{2},\lambda \dots \Rightarrow y_{sw}=2a=$ Antinode
At $x=\frac{\lambda }{4},\frac{3\lambda }{4}..\Rightarrow y_{sw}=0=$ Nodes