Consider a system of three connected strings, S₁, S₂ and S₃ with uniform linear mass densities \,{kg/m}, 4\,{kg/m} and 16\,{kg/m}, respectively, as shown in the figure. S₁ and S₂ are connected at the point P, whereas S₂ and S₃ are connected at the point Q, and the other end of S₃ is connected to a wall. A wave generator O is connected to the free end of S₁. The wave from the generator is represented by y=y₀ (ω t - k x)\,{cm}, where y₀, ω and k are constants of appropriate dimensions. Which of the following statements is/are correct: [JEE Adv 2025 P1]

When the wave reflects from P for the first time, the reflected wave is represented by y=α₁ y₀ (ω t + k x + π)\,{cm}, where α₁ is a positive constant

When the wave transmits through P for the first time, the transmitted wave is represented by y=α₂ y₀ (ω t - k x) cm , where α₂ is a positive constant

When the wave reflects from Q for the first time, the reflected wave is represented by y=α₃ y₀ (ω t - k x + π)\,{cm}, where α₃ is a positive constant

When the wave transmits through Q for the first time, the transmitted wave is represented by y=α₄ y₀ (ω t - 4 k x) cm , where α₄ is a positive constant

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JEE Advanced 2025 Paper 1

Section: Physics

Question : 23 of 48

Marks: +1, -0

Consider a system of three connected strings,

S_1, S_2

and

S_3

with uniform linear mass densities

\mu\,\text{kg/m}, 4\mu\,\text{kg/m}

and

16\mu\,\text{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 (\omega t - k x)\,\text{cm}

, where

y_0

\omega

and

k

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

[JEE Adv 2025 P1]

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

Validate

Solution:

Reflection of wave in string from fixed end, reflected wave phase change by

\pi

And reflected wave from free (rarer) end No phase change.

\because\; \omega = k V

\omega = k \sqrt{ \frac{T}{\mu} }\;\; \text{given}\;\; y=y_0 \cos (\omega t - k x)\,\text{cm}

\frac{\sqrt{T}}{\mu} = \text{ const. }\;\;\; \text{ For medium } P Q (S_2)

k_{S_2}=k_{PQ}=2k

\text{ For medium } S_3

k_{S_3}=4k

Option (A) reflection from

P

, phase change by

\pi

No change in '

k

'. Hence correct option.

In option (B)

k_{s_2}=2k

but in option given

k_s=k

Hence wrong.

In option (C) direction of reflected wave in negative direction.

Hence wrong.

Option (D) no phase change in transmitted wave and given

k_{s_3}=4k

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