Waves

JEE Main level, single correct, medium

56 of 142

Question 56 of 142

A wave represented by the equation y1=acos⁡y_1=a \cos (kx−ωt)(k x-\omega t) is superimposed with another wave to form a stationary wave such that the point x−0x-0 is node. The equation for the other wave is

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Answer: (B) acos⁡(kx+ωt+π)a \cos (k x+\omega t+\pi)

Since the point x=0x=0 is a node and reflection is taking place from point x=0x=0. This means that reflection must be taking place from the fixed end and hence the reflected ray must suffer an additional phase change of π\pi or a path change of λ2\frac{\lambda}{2}.
So, if yincident =acos⁡(kx−ωt)y_{\text {incident }}=a \cos (k x-\omega t)

⇒yincident =acos⁡(−kx−ωt+π)=−acos⁡(ωt+kx)\begin{aligned} \Rightarrow y_{\text {incident }} & =a \cos (-k x-\omega t+\pi) \\ & =-a \cos (\omega t+k x) \end{aligned}
Hence equation for the other wave
y=acos⁡(kx+ωt+π)y=a \cos (k x+\omega t+\pi)

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