The two nearest harmonics of a tube closed at one end and open at other end are 220 Hz and 260Hz. What is the fundamental frequency of the system?

We know, for a tube closed at one end, frequency n is an odd positive integer
where v is the velocity of sum and $\ell$ is the length of the air column.

$f=\frac{nv}{4\ell }$

Given, two nearest frequencies

${\mathrm{f}}_1=220\mathrm{Hz},{\mathrm{f}}_2=260\mathrm{Hz}$

${\mathrm{f}}_1=\frac{\mathrm{nv}}{4\ell }=220\mathrm{Hz} \text{and }{\mathrm{f}}_2=\frac{(\mathrm{n}+2)\mathrm{v}}{4\ell }=260\mathrm{Hz}$

$\Rightarrow {\mathrm{f}}_2-{\mathrm{f}}_1=\frac{2\mathrm{v}}{4\ell }=40\mathrm{Hz}$

$\frac{\mathrm{v}}{2\ell }=40\mathrm{Hz}$

Fundamental frequency is given by $\frac{\mathrm{v}}{4\ell }$

$\therefore \mathbf{f}=20\mathrm{Hz}$