Two open organ pipes of fundamental frequenci | vedclass

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(B) The fundamental frequency of an open organ pipe of length $\ell$ is given by $n = \frac{V}{2\ell}$, where $V$ is the speed of sound.For the two pi

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$\frac{n_1 n_2}{n_1 + n_2}$

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(B) The fundamental frequency of an open organ pipe of length $\ell$ is given by $n = \frac{V}{2\ell}$, where $V$ is the speed of sound.For the two pipes, we have $n_1 = \frac{V}{2\ell_1}$ and $n_2 = \frac{V}{2\ell_2}$.This implies $\ell_1 = \frac{V}{2n_1}$ and $\ell_2 = \frac{V}{2n_2}$.When the two pipes are joined in series, the total length of the new pipe is $\ell = \ell_1 + \ell_2$.The fundamental frequency $n$ of the new pipe is $n = \frac{V}{2\ell} = \frac{V}{2(\ell_1 + \ell_2)}$.Substituting the values of $\ell_1$ and $\ell_2$, we get $n = \frac{V}{2(\frac{V}{2n_1} + \frac{V}{2n_2})} = \frac{V}{V(\frac{1}{n_1} + \frac{1}{n_2})} = \frac{1}{\frac{n_1 + n_2}{n_1 n_2}} = \frac{n_1 n_2}{n_1 + n_2}$.

Two open organ pipes of fundamental frequencies $n_1$ and $n_2$ are joined in series. The fundamental frequency of the new pipe so obtained will be

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