JEE Advanced 2025 · Paper 1 · Physics Q7

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PhysicsMultiple correct+4 / -2Question 23 of 48

Consider a system of three connected strings, $S_1$ , $S_2$ and $S_3$ with uniform linear mass densities $\mu$ kg/m, $4\mu$ kg/m and $16\mu$ 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 $y = y_0\cos(\omega t - kx)$ cm, where $y_0$ , $\omega$ and $k$ are constants of appropriate dimensions. Which of the following statements is/are correct:

Axes: X to the right, Y up. A horizontal string runs from the wave generator O on the left to a wall on the right: S₁ (linear density μ) from O to point P, S₂ (4μ) from P to Q, and S₃ (16μ) from Q to the wall. The strings get thicker/darker from S₁ to S₃.

When the wave reflects from

P

for the first time, the reflected wave is represented by

y = \alpha_1y_0\cos(\omega t + kx + \pi)

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_2y_0\cos(\omega t - kx)

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_3y_0\cos(\omega t - kx + \pi)

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_4y_0\cos(\omega t - 4kx)

cm, where

\alpha_4

is a positive constant.

Answer and worked solution

Official answer: A, D

Speeds and wave numbers. Wave speed is $\sqrt{T/\mu}$ and the frequency is unchanged, so $k\propto\sqrt\mu$ . The wave numbers are $k$ , $2k$ and $4k$ in $S_1$ , $S_2$ and $S_3$ .

At P.

Reflection is from a denser string, so the wave inverts and travels in $-x$ : $y=\alpha_1y_0\cos(\omega t+kx+\pi)$ . (A) ✓
The transmitted wave has wave number $2k$ : $\cos(\omega t-2kx)$ , not $\cos(\omega t-kx)$ . (B) ✗

At Q.

The reflected wave travels back in $S_2$ with wave number $2k$ , inverted: $\cos(\omega t+2kx+\pi)$ , not $\cos(\omega t-kx+\pi)$ . (C) ✗
The transmitted wave in $S_3$ has wave number $4k$ : $\cos(\omega t-4kx)$ . (D) ✓

Topics:Wave equation and progressive waves Speed of waves

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JEE Advanced 2025 · Paper 1, Physics question 7