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

Correct answer: A. When the wave reflects from P for the first time, the reflected wave is represented by y=α_1 y_0 (ω t+k x+π) cm , where α_1 is a positive constant. (A) y_1=y_0 (ω t-k x) when wave going from Rarer to Denser, aligned & y_r=A_r (ω t+k x+π) & y_r=a_1 y_0 (ω t+k x+π) aligned option (A) correct (B) For transmitted from point P aligned & y_ t =A_ t [ω t-k_1 x] & k_1 k = (μ_1)/(μ) = k_1 k = (4 μ)/(μ) & k_1=2 k & y_ t =a_2 y_0 [ω t-2 kx] aligned option (B) incorrect (C) when reflected from Q aligned & y_i=a_2 y_0 [ω t-2 k x] & y_r=a_3 y_0 [ω t+2 k x+π] aligned option (C) incorrect (D) when transmitted from Q aligned & y_ t =a_4 y_0 [ω t=k_2 x] & k_2 2 k = (16 μ)/(4 μ) k_2=4 k & y_ t =a_4 y_0 [ω t-4 kx] aligned option (D) correct

JEE Advanced 2025Paper 1

Exam: JEE Advanced
Year: 2025
Paper: Paper 1
Subject: Physics
Chapter: Waves and Sound

JEE Advanced 2025 · Paper 1 · Physics · Waves and Sound

Consider a system of three connected strings, S_1, S_2 and S_3 with...

Consider a system of three connected strings,

S_1, S_2

and

S_3

with uniform linear mass densities

\mu \mathrm{kg} / \mathrm{m}

4 \mu \mathrm{~kg} / \mathrm{m}

and

16 \mu \mathrm{~kg} / \mathrm{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 (\omega t-k x) \mathrm{cm}

, where

y_0, \omega

and

k

are constants of appropriate dimensions. Which of the following statements is/are correct:
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 /

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 /

Options

AWhen the wave reflects from $P$ for the first time, the reflected wave is represented by $y=\alpha_1 \mathrm{y}_0 \cos (\omega t+k x+\pi) \mathrm{cm}$ , where $\alpha_1$ is a positive constant.
BWhen the wave transmits through $P$ for the first time, the transmitted wave is represented by $y=\alpha_2 \mathrm{y}_0 \cos (\omega t-k x) \mathrm{cm}$ , where $\alpha_2$ is a positive constant.
CWhen the wave reflects from $Q$ for the first time, the reflected wave is represented by $y=\alpha_3 y_0 \cos (\omega t-k x+\pi) \mathrm{cm}$ , where $\alpha_3$ is a positive constant.
DWhen the wave transmits through $Q$ for the first time, the transmitted wave is represented by $y=\alpha_4 y_0 \cos (\omega t-4 k x) \mathrm{cm}$ , where $\alpha_4$ is a positive constant.

Correct answer

A. When the wave reflects from $P$ for the first time, the reflected wave is represented by $y=\alpha_1 \mathrm{y}_0 \cos (\omega t+k x+\pi) \mathrm{cm}$ , where $\alpha_1$ is a positive constant.

Step-by-step solution

(A)

y_1=y_0 \cos (\omega t-k x)

when wave going from Rarer to Denser,

\begin{aligned} & y_r=A_r \cos (\omega t+k x+\pi) \\ & y_r=a_1 y_0 \cos (\omega t+k x+\pi)\end{aligned}

option (A) correct (B) For transmitted from point P

\begin{aligned} & \mathrm{y}_{\mathrm{t}}=\mathrm{A}_{\mathrm{t}} \cos \left[\omega \mathrm{t}-\mathrm{k}_1 \mathrm{x}\right] \\ & \frac{\mathrm{k}_1}{\mathrm{k}}=\sqrt{\frac{\mu_1}{\mu}}=\frac{\mathrm{k}_1}{\mathrm{k}}=\sqrt{\frac{4 \mu}{\mu}} \\ & \mathrm{k}_1=2 \mathrm{k} \\ & \mathrm{y}_{\mathrm{t}}=\mathrm{a}_2 \mathrm{y}_0 \cos [\omega \mathrm{t}-2 \mathrm{kx}]\end{aligned}

option (B) incorrect (C) when reflected from Q

\begin{aligned} & y_i=a_2 y_0 \cos [\omega t-2 k x] \\ & y_r=a_3 y_0 \cos [\omega t+2 k x+\pi]\end{aligned}

option (C) incorrect (D) when transmitted from Q

\begin{aligned} & y_{\mathrm{t}}=a_4 y_0 \cos \left[\omega \mathrm{t}=k_2 \mathrm{x}\right] \\ & \frac{k_2}{2 \mathrm{k}}=\sqrt{\frac{16 \mu}{4 \mu}} \Rightarrow k_2=4 \mathrm{k} \\ & y_{\mathrm{t}}=a_4 y_0 \cos [\omega \mathrm{t}-4 \mathrm{kx}]\end{aligned}

option (D) correct

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