A sinusoidal wave represented by $y_{i} = 0.3 cos (2 x - 40 t)$ is travelling along a string towards a fixed end at $x = 0$ . Then, the equation of the reflected wave will be:-
(Assume there is no loss of energy at the boundary)

0.3 cos (2 x + 40 t + \frac{π}{2})

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0.3 cos (2 x - 40 t + \frac{π}{2})

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0.3 cos (2 x - 40 t + π)

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0.3 cos (2 x + 40 t - π)

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Solution

The correct option is D $0.3 cos (2 x + 40 t - π)$
Given that,
Equation of incident wave $y_{i} = 0.3 cos (2 x - 40 t)$

Since the wave is moving towards a fixed end $x = 0$ , we can say that, $A_{t} = 0, A_{r} = - A_{i}$

After reflection, the wave moves in the direction opposite to the direction of incident wave.
Therefore, equation of the reflected wave will be
$y_{r} = - 0.3 cos (2 x + 40 t)$
$∵ cos (π - θ) = - cos θ$ we get ,
$y_{r} = 0.3 cos (π - (2 x + 40 t))$
or $y_{r} = 0.3 cos (2 x + 40 t - π)$
[since $cos (- x) = cos x$ ]
Hence, option (d) is the correct answer.

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Reflection from a Fixed End

Standard XII Physics

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A sinusoidal wave represented by yi=0.3cos(2x−40t) is travelling along a string towards a fixed end at x=0. Then, the equation of the reflected wave will be:- (Assume there is no loss of energy at the boundary)

A sinusoidal wave represented by $y_{i} = 0.3 cos (2 x - 40 t)$ is travelling along a string towards a fixed end at $x = 0$ . Then, the equation of the reflected wave will be:-
(Assume there is no loss of energy at the boundary)