A source, approaching with speed $u$ towards the open end of a stationary pipe of length $L$, is emitting a sound of frequency $f_5$. 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 $f_5$, will the sound reaching the pipe lead to a resonance?
$(A)$ $u=0.8 v$ and $f_5=f_0$
$(B)$ $u=0.8 v$ and $f_5=2 f_0$
$(C)$ $u=0.8 v$ and $f_5=0.5 f_0$
$(D)$ $u=0.5 v$ and $f_5=1.5 f_0$
IIT 2021, Advanced
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$f=f\left(\frac{v}{v-u}\right)$
$(A)$ $f=f_0\left(\frac{v}{v-0.8 v}\right)=5 f_0$
$(B)$ $f =2 f _0\left(\frac{ v }{ v -0.8 v }\right)=10 f _0$
$(C)$ $f =0.5 f _0\left(\frac{ v }{ v -0.8 v }\right)=2.5 f _0$
$(D)$ $f =1.5 f _0\left(\frac{ v }{ v -0.5 v }\right)=3 f _0$
All odd harmonics are available in closed pipe therefore correct Ans $(A,D)$
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