MHT CET · Physics · Waves and Sound

A wave travelling along uniform string represented by \(Y=A \sin (\omega t-k x)\) is superimposed on another wave travelling along the same string represented by \(Y=A \sin (\omega t+k x)\). The resultant is

A A wave travelling along \(+x\) direction.
B A standing wave having nodes at \(x=\left(n+\frac{1}{2}\right) \frac{\lambda}{2}\), where \(n=0,1,2,3 \ldots \ldots\)
C A wave travelling along \(-x\) direction
D A standing wave having nodes at \(x=\frac{n \lambda}{2}\), where \(n=0,1,2,3 \ldots \ldots\).

Verified Solution

Answer & Solution

Correct Answer

(B) A standing wave having nodes at \(x=\left(n+\frac{1}{2}\right) \frac{\lambda}{2}\), where \(n=0,1,2,3 \ldots \ldots\)

Step-by-step Solution

Detailed explanation

The equation of the standing wave formed due to superposition of \(Y=A \sin (\omega t-k x)\) and \(Y=A \sin (\omega t+k x)\) is given by,
\(Y=[2 A \cos (k x)] \sin (\omega t)\)
Location of the nodes is dictated by condition:
\([2 A \cos (k x)]=0\)
Therefore,
\(\begin{aligned} & k x=\left(\frac{2 \pi}{\lambda}\right) x=(2 n+1) \frac{\pi}{2} \\ & \Rightarrow x=\left(n+\frac{1}{2}\right) \frac{\lambda}{2} ; n=0,1,2, \ldots .\end{aligned}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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