Consider a system of three connected strings, , and with uniform linear mass densities , and , respectively, as shown in the figure. and are connected at the point , whereas and are connected at the point , and the other end of is connected to a w

Question

Consider a system of three connected strings, $S_{1}$ , $S_{2}$ and $S_{3}$ with uniform linear mass densities $μ kg/m$ , $4 μ kg/m$ and $16 μ kg/m$ , respectively, as shown in the figure. $S_{1}$ and $S_{2}$ are connected at the point $P$ , whereas $S_{2}$ and $S_{3}$ are connected at the point $Q$ , and the other end of $S_{3}$ is connected to a wall. A wave generator $O$ is connected to the free end of $S_{1}$ . The wave from the generator is represented by $y = y_{0} \cos (ω t - k x) cm,$ where $y_{0}$ , $ω$ and $k$ are constants of appropriate dimensions. Which of the following statements is/are correct: [2025]

Smart Achievers · Accepted Answer

(1, 4)

When wave going from rarer to denser, phase change by $π$

$y_{i} = y_{0} \cos (ω t - k x)$

$y_{r} = A_{r} \cos (ω t + k x + π)$

$Q$

So, option (1) correct.

When transmitted from point P,

$y_{t} = A_{t} \cos [ω t - k_{1} x]$

$\frac{k_{1}}{k} = \sqrt{\frac{μ_{1}}{μ}} = \frac{k_{1}}{k} = \sqrt{\frac{4 μ}{μ}}$

$k_{1} = 2 k$

$y_{t} = α_{2} y_{0} \cos [ω t - 2 k x]$

So, option (2) incorrect.

When reflected from $Q$ ,

$y_{i} = a_{2} y_{0} \cos [ω t - 2 k x]$

$y_{r} = a_{3} y_{0} \cos [ω t + 2 k x + π]$

So, option (3) incorrect.

When transmitted from $Q$ ,

$y_{t} = α_{4} y_{0} \cos [ω t - k_{2} x]$

$\frac{k_{2}}{2 k} = \sqrt{\frac{16 μ}{4 μ}}$

$k_{2} = 4 k$

$y_{t} = α_{4} y_{0} \cos [ω t - 4 k x]$

So, option (4) correct.

Solution

1 When the wave reflects from $P$ for the first time, the reflected wave is represented by $y = α_{1} y_{0} \cos (ω t + k x + π) cm,$ where $α_{1}$ is a positive constant.

2 When the wave transmits through $P$ for the first time, the transmitted wave is represented by $y = α_{2} y_{0} \cos (ω t - k x) cm,$ where $α_{2}$ is a positive constant.

3 When the wave reflects from $Q$ for the first time, the reflected wave is represented by $y = α_{3} y_{0} \cos (ω t - k x + π) cm,$ where $α_{3}$ is a positive constant.

4 When the wave transmits through $Q$ for the first time, the transmitted wave is represented by $y = α_{4} y_{0} \cos (ω t - 4 k x) cm,$ where $α_{4}$ is a positive constant.

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Solution

1 When the wave reflects from P for the first time, the reflected wave is represented by y=α1y0cos(ωt+kx+π) cm, where α1 is a positive constant.

2 When the wave transmits through P for the first time, the transmitted wave is represented by y=α2y0cos(ωt-kx) cm, where α2 is a positive constant.

3 When the wave reflects from Q for the first time, the reflected wave is represented by y=α3y0cos(ωt-kx+π) cm, where α3 is a positive constant.

4 When the wave transmits through Q for the first time, the transmitted wave is represented by y=α4y0cos(ωt-4kx) cm, where α4 is a positive constant.

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1 When the wave reflects from $P$ for the first time, the reflected wave is represented by $y = α_{1} y_{0} \cos (ω t + k x + π) cm,$ where $α_{1}$ is a positive constant.

2 When the wave transmits through $P$ for the first time, the transmitted wave is represented by $y = α_{2} y_{0} \cos (ω t - k x) cm,$ where $α_{2}$ is a positive constant.

3 When the wave reflects from $Q$ for the first time, the reflected wave is represented by $y = α_{3} y_{0} \cos (ω t - k x + π) cm,$ where $α_{3}$ is a positive constant.

4 When the wave transmits through $Q$ for the first time, the transmitted wave is represented by $y = α_{4} y_{0} \cos (ω t - 4 k x) cm,$ where $α_{4}$ is a positive constant.