A pipe $30\, cm$ long, is open at both ends. Which harmonic mode of the pipe resonates a $1.1\, kHz$ source ? (Speed of sound in air $= 330\, ms^{-1}$)
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Here,
Speed of sound, $\mathrm{v}=330 \mathrm{\,ms}^{-1}$
Length of pipe, $\mathrm{L}=30 \mathrm{\,cm}=30 \times 10^{-2} \mathrm{\,m}$
In a open pipe (open at both ends), the frequency of its $n^{\text {th }}$ harmonic is
$v_{n}=\frac{n v}{2 L}$ where $n=1,2,3........$
$\therefore \mathrm{n}=\frac{2 \mathrm{L} v_{\mathrm{n}}}{\mathrm{v}}$
Let $n^{\text {th }}$ harmonic of open pipe resonate with $1.1$ $\mathrm{\,kHz}$ source.
$\therefore \mathrm{v}_{\mathrm{n}}=1.1 \mathrm{\,kHz}=1.1 \times 10^{3} \mathrm{\,Hz}$
$\therefore \mathrm{n}=\frac{2 \times 30 \times 10^{-2} \times 1.1 \times 10^{3}}{330}=2$
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