JEE Advanced 2025 — Paper 1 — PHY-II solution

JEE Advanced2025Paper 1PHY-II

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:

A
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.
B
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.
C
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.
D
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.

Solution

Wave speed on a string under tension $T$ is $v = T / μ$ . With identical tension throughout, $v \propto 1/ μ$ , so the wave number scales as $k \propto μ$ .

At interface P (going from $μ$ to $4 μ$ , rarer to denser):

$\frac{k _{1}}{k} = \frac{4 μ}{μ} = 2 \Rightarrow k_{1} = 2 k$

Reflection from a denser medium introduces a phase shift of $π$ , so the reflected wave is $y_{r} = α_{1} y_{0} cos (ω t + k x + π)$ . Option (A) is correct.

The transmitted wave in $S_{2}$ has wave number $2 k$ : $y_{t} = α_{2} y_{0} cos (ω t - 2 k x)$ , not $cos (ω t - k x)$ . Option (B) is incorrect.

At interface Q (going from $4 μ$ to $16 μ$ , again rarer to denser):

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

The incident wave at Q is $α_{2} y_{0} cos (ω t - 2 k x)$ . The reflected wave in $S_{2}$ is $α_{3} y_{0} cos (ω t + 2 k x + π)$ , not $cos (ω t - k x + π)$ . Option (C) is incorrect.

The transmitted wave in $S_{3}$ has wave number $4 k$ : $y_{t} = α_{4} y_{0} cos (ω t - 4 k x)$ . Option (D) is correct.

$∴$ Correct options are (A) and (D).

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