Question

An open pipe $40\text{cm}$ long and a closed pipe $31\text{cm}$ long, both having same diameter, are producing their first overtone, and these are in unison. Determine the end correction(in $\text{cm}$) of these pipes.

Answer 1

Let $v$ be the velocity of sound and $f$ be the frequency of the note.
Thus, the wavelength of sound produced is $\lambda =\frac{v}{f}$
The wavelength of first overtone of open organ pipe is
${\lambda }_o=l+2e=40+2e\Rightarrow \frac{v}{f}=40+2e$
The wavelength of first overtone of closed organ pipe is
${\lambda }_c=\frac{4}{3}(l+e)=\frac{4}{3}(31+e)\Rightarrow \frac{v}{f}=\frac{4}{3}(31+e)$

By eliminating $\frac{v}{f}$ from the above two equations, we get
$\frac{3}{4}(40+2e)=31+e\Rightarrow 120+6e=124+4e\Rightarrow e=2\text{cm}$