Question

The fundamental frequency of an air column in a pipe closed at one end is in unison with the third overtone of an open pipe. Calculate the ratio of lengths of their air columns.

Answer 1

Given that $n_3=n_0$

were $n_3=$ frequency of the third overtone of the open pipe

$n_0=$ fundamental frequency of the closed pipe

Third overtone of open pipe is

$n_3=4\left(\frac{V}{2L_3}\right)$

Fundamental frequency of closed pipe at one end is.

$n_0=\frac{V}{4L_0}$

$\frac{V}{4L_0}=4\left(\frac{V}{2L_3}\right)$ $(\because n_0=n_3)$

$\therefore \frac{L_0}{L_3}=\frac{1}{8}$

$\therefore L_0:L_3=1:8$