A source, approaching with speed towards the open end of a stationary pipe of length , is emitting a sound of frequency . The farther end of the pipe is closed. The speed of sound in air is and is the fundamental frequency of the pipe. For which of the

Question

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? [2021]

Smart Achievers · Accepted Answer

(1, 4)

$From Doppler's effect$

$f_{s}$

$As the pipe is closed at one end$ $∴$ $For resonance,$

$f = f_{0} (\frac{v}{v - u}) = n f_{0}$ where $n =$ odd integer

$(1) For, u = 0.8 v and f_{s} = f_{0},$

$f = f_{0} (\frac{v}{v - 0.8 v}) = 5 f_{0}$

$(2) For, u = 0.8 v and f_{s} = 2 f_{0}$ $\Rightarrow f = 2 f_{0} (\frac{v}{v - 0.8 v}) = 10 f_{0}$

$(3) For, u = 0.8 v and f_{s} = 0.5 f_{0}$

$f = 0.5 f_{0} (\frac{v}{v - 0.8 v}) = 2.5 f_{0}$

$(4) For u = 0.5 v and f_{s} = 1.5 f_{0}$

$f = 1.5 f_{0} (\frac{v}{v - 0.5 v}) = 3 f_{0}$

Solution

1 $u = 0.8 v$ and $f_{s} = f_{0}$

2 $u = 0.8 v$ and $f_{s} = 2 f_{0}$

3 $u = 0.8 v$ and $f_{s} = 0.5 f_{0}$

4 $u = 0.5 v$ and $f_{s} = 1.5 f_{0}$

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Solution

1 u=0.8v and fs=f0

2 u=0.8v and fs=2f0

3 u=0.8v and fs=0.5f0

4 u=0.5v and fs=1.5f0

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1 $u = 0.8 v$ and $f_{s} = f_{0}$

2 $u = 0.8 v$ and $f_{s} = 2 f_{0}$

3 $u = 0.8 v$ and $f_{s} = 0.5 f_{0}$

4 $u = 0.5 v$ and $f_{s} = 1.5 f_{0}$