Question

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:

• A. $n_1+n_2$
• B. $\frac{n_1{n}_2}{n_1+n_2}$
• C. $n_1+\frac{n_2}{2}$
• D. None of the above

Answer 1

The correct option is B $\frac{n_1{n}_2}{n_1+n_2}$

Fundamental frequency of an open pipe of length $L$ is given by $n=\frac{v}{2L}$
$⟹$ $L=\frac{v}{2n}$
So, we get lengths of two open pipes as $L_1=\frac{v}{2n_1}$ and $L_2=\frac{v}{2n_2}$
Now the pipes are join in series.
Fundamental frequency of new open pipe of length $L_1+L_2$, $n=\frac{v}{2(L_1+L_2)}$
$\therefore$ $n=\frac{v}{2(\frac{v}{2n_1}+\frac{v}{2n_2})}$
Or $n=\frac{1}{\frac{1}{n_1}+\frac{1}{n_2}}$
$⟹n=\frac{n_1{n}_2}{n_1+n_2}$