Question
Prove that a pipe of length $2 L$ open at both ends has the same fundamental frequency as a pipe of length $L$ closed at one end.
✓
Answer
Let $L_o$ and $L_c$ be the lengths of a pipe open at both ends and a pipe closed at one end, respectively. Let $n_{\circ}$ and $n_c$ be their corresponding fundamental frequencies. Then, ignoring the end corrections,
$
n _{ O }=\frac{v}{2 L_0} \text { and } n _{ C }=\frac{v}{4 L_{ C }}
$
where $v$ is the speed of the sound in air.
Given that $L_c=L_{\text {and }} L_0=2 L$,
$n _{ O }=\frac{v}{4 L}$ and $n _{ C }=\frac{v}{4 L} \therefore n _{ O }= n _{ C }$
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