JEE Advanced 2021 Paper 2 Q02 Physics Waves & Optics Sound Waves Medium

JEE Advanced 2021 Paper 2 · Q02 · Sound Waves

A source, approaching with speed $u$ towards the open end of a stationary pipe of length $L$, is emitting a sound of frequency $f_s$. The farther end of the pipe is closed. The speed of sound in air is $v$ and $f_0$ is the fundamental frequency of the pipe. For which of the following combination(s) of $u$ and $f_s$, will the sound reaching the pipe lead to a resonance?

  1. A. $u = 0.8 v$ and $f_s = f_0$
  2. B. $u = 0.8 v$ and $f_s = 2 f_0$
  3. C. $u = 0.8 v$ and $f_s = 0.5 f_0$
  4. D. $u = 0.5 v$ and $f_s = 1.5 f_0$
JEE Advanced multi-correct — pick every correct option, then check.
Reveal answer + step-by-step solution

Correct answer:A, D

Solution

A pipe closed at one end resonates at frequencies $(2n-1) f_0$ for $n = 1, 2, 3, \ldots$ (i.e. $f_0, 3f_0, 5f_0, \ldots$). The source approaches the pipe, so the apparent frequency at the open end is $f_{\text{app}} = f_s \cdot v/(v - u)$. (A) $f_{\text{app}} = f_0 \cdot v/(0.2v) = 5 f_0$ — matches $(2n-1)f_0$ with $n=3$, resonance. (B) $f_{\text{app}} = 2f_0 \cdot 5 = 10 f_0$ — even multiple, no. (C) $f_{\text{app}} = 0.5 f_0 \cdot 5 = 2.5 f_0$ — no. (D) $f_{\text{app}} = 1.5 f_0 \cdot v/(0.5v) = 3 f_0$ — matches $n=2$, resonance.

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