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A travelling wave $Y = A\sin \left( {kx - \omega t + \theta } \right)$ passes from a heavier string to lighter string. The reflected has amplitude $0 \cdot 5{\text{A}}$. The junction of the string is $x = 0$. The equation of the reflected wave is:
A) $y' = 0 \cdot 5A\sin \left( {kx + \omega t + \theta } \right)$
B) $y' = - 0 \cdot 5A\sin \left( {kx + \omega t + \theta } \right)$
C) $y' = - 0 \cdot 5A\sin \left( {kx - \omega t - \theta } \right)$
D) $y' = - 0 \cdot 5A\sin \left( {kx + \omega t - \theta } \right)$

seo-qna
Last updated date: 20th Jun 2024
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Answer
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Hint: A simple harmonic motion is a periodic motion along two extremes and an equilibrium position. The wave has some amplitude which is the maximum displacement from the mean position and phase represents the relationship between the waves with same frequency if the phase of the two waves is not same then the two waves will not move together they will have some difference while moving there will be some fixed distance between their crest and trough of the two waves.

Complete step by step solution:
It is given that the wave $Y = A\sin \left( {kx - \omega t + \theta } \right)$ is travelling from a heavier string to the lighter string and we need to find the equation of the wave which is reflected back in the heavier string. The change of the phase is only in the reflected wave the phase of the reflected wave increases by$\pi $.
When the wave is travelling from the heavier string to the lighter string then there is no change in the phase of the wave but if the wave gets reflected back in the same medium then there is change in the phase of the reflected wave. The reflected wave in the same medium is given by,
$ \Rightarrow y' = 0 \cdot 5A\sin \left( {kx + \omega t + \theta + \pi } \right)$
After solving we get,
$ \Rightarrow y' = - 0 \cdot 5A\sin \left( {kx + \omega t + \theta } \right)$
So the equation of the reflected wave is given by $y' = - 0 \cdot 5A\sin \left( {kx + \omega t + \theta } \right)$.

The correct option for this problem is option B.

Note: Whenever there is change of medium in the travelling of the wave then there are two phenomena that take place one is transverse and other is reflection. The reflection happens to the same medium and the transverse is in the other medium and there is no change of phase in the transverse.